SPUR  AND    BEVEL   GEARING 


SPUR  AND  BEVEL 
GEARING 


A  TREATISE  ON  THE  PRINCIPLES,  DIMEN- 
SIONS, CALCULATION,  DESIGN  AND  STRENGTH 
OF  SPUR  AND  BEVEL  GEARING,  TOGETHER 
WITH  CHAPTERS  ON  SPECIAL  TOOTH  FORMS 
AND  METHODS  OF  CUTTING  GEAR  TEETH 


COMPILED  AND  EDITED 
BY 

ERIK   OBERG,   A.  S.  M.  E. 

ASSOCIATE  EDITOR  OF  MACHINERY 
EDITOR  OF  MACHINERY'S  HANDBOOK 

AUTHOR  OF  "  HANDBOOK  OF  SMALL  TOOLS,"  "  SHOP  ARITHMETIC  FOR  THE  MACHINIST," 

"  SOLUTION  OF  TRIANGLES,"  "  STRENGTH  OF  MATERIALS," 

"ELEMENTARY  ALGEBRA,"  ETC. 


FIRST  EDITION 

FIF7?H,  PRINTING 


NEW  YORK 

THE    INDUSTRIAL   PRESS 

LONDON  :    THE  MACHINERY  PUBLISHING  CO.,  LTD. 
1920 


COPYRIGHT,  1914 

BY 

THE  INDUSTRIAL  PRESS 
NEW  YORK 


PREFACE 


EARLY  in  1908,  MACHINERY  began  the  publication  of  its 
well-known  25-cent  Reference  Books.  These  include  the  best 
of  the  material  that  has  appeared  in  MACHINERY  during  the 
past  twenty  years,  adequately  revised,  amplified  and  brought 
up-to-date.  Of  these  books,  one  hundred  and  twenty-five  dif- 
ferent titles  have  been  published  in  the  past  six  years. 

Many  subjects,  however,  cannot  be  covered  in  all  their  phases 
to  an  adequate  extent  in  books  of  this  size,  and  in  answer  to  a 
demand  for  more  comprehensive  and  detailed  treatments  on  the 
more  important  mechanical  subjects,  it  has  been  deemed  advis- 
able to  bring  out  a  number  of  larger  volumes,  each  covering 
one  subject  completely.  This  book  is  one  of  these  volumes. 
In  bringing  out  these  books,  the  first  consideration  on  the  part 
of  the  editors  has  been  to  make  them  meet  the  practical  needs 
of  the  machine  building  trade.  Mere  theory  and  academic  dis- 
cussions have  been  avoided.  The  rules,  formulas  and  instruc- 
tions are  illustrated  with  engravings  whenever  practicable,  and 
examples  are  given  to  show  their  application  to  every-day 
problems.  Theoretical  considerations,  however,  have  not  been 
neglected,  when  necessary  to  fully  explain  a  practical  process, 
and  the  present  book  on  spur  and  bevel  gearing  is,  therefore,  a 
treatise  on  both  the  practice  and  the  theory  of  gearing,  along 
such  lines  as  will  make  it  especially  useful  to  practical  men. 

The  information  contained  is  mainly  compiled  from  articles 
published  in  MACHINERY,  and  the  best  on  the  subject  that 
has  appeared  in  the  Reference  Books  is  also  included.  Ampli- 
fications and  additions  have  been  made  wherever  necessary. 
For  the  material  contained,  MACHINERY  is  indebted  to  a  large 
number  of  men  who  have  furnished  practical  information  to 
its  columns  in  the  past.  In  many  instances,  it  has  not  been 


416066 


VI  PREFACE 

possible  to  give  credit  to  each  individual  contributor,  but  it 
should  be  mentioned  that  the  framework  upon  which  the  whole 
book  has  been  built  up,  consists  of  the  Reference  Books  and 
articles  which  Mr.  Ralph  E.  Flanders,  the  well-known  gear 
expert  and  formerly  Associate  Editor  of  MACHINERY,  has 
written  and  compiled.  To  all  other  writers  whose  material 
has  appeared  in  MACHINERY  and  is  now  used  in  this  book, 
MACHINERY  hereby  expresses  its  appreciation. 

MACHINERY 

NEW  YORK,  May,  1914 


CONTENTS 


CHAPTER  I 
PRINCIPLES  AND  DIMENSIONS  OF  SPUR  GEARING 

PAGES 

Friction  Wheels  —  Toothed  Gearing  —  Involute  and  Cy- 
cloidal  Systems  —  Pressure  Angles  —  Definitions  of  Gear- tooth 
Parts  —  Circular  and  Diametral  Pitch  —  Pitch  Diameter  — 
Center  Distance  —  Clearance  —  Depth  and  Thickness  of 
Tooth  —  Outside  Diameter  —  Number  of  Teeth  —  Table  of 
Rules  and  Formulas  —  Internal  Spur  Gears  —  Grant's 
Odontograph  —  Cutters  for  Milling  Gear  Teeth  —  Chordal 
Thickness  and  Corrected  Addendum  —  Testing  Tooth  Thick- 
ness —  Limits  for  Gearing  —  Metric  System  of  Gear  Teeth. . .  1-37 

CHAPTER  II 
MATERIALS  USED   FOR  GEARS 

Cast  Iron  —  Cast  Steel  —  Materials  for  Medium-sized  and 
Small  Gears  —  Brass  and  Bronze  Gears  —  Rawhide  Gears  — 
Fiber  Gears  —  Cloth  Gears  —  Materials  for  Racks  —  Gears 
for  Machine  Tool  Drives  —  Safe  Tooth  Pressures  for  Cast- 
iron  Gears  —  Grades  of  Steel  giving  Best  Results  —  Heat- 
treatment  of  Gears 38-50 

CHAPTER  III 
STRENGTH  AND   DURABILITY  OF  SPUR  GEARING 

Strength  of  Gear  Teeth  —  The  Lewis  Formula  —  Relation 
between  Width  of  Face  and  Diametral  Pitch  —  Derivation 
of  the  Lewis  Formula  —  Tests  for  Strength  of  Gear  Teeth  — 
Rawhide  Gearing  —  Strength  of  Rawhide  Gears  with  Flanges 
—  Durability  of  Gearing  —  Efficiency  of  Standard  Spur 
Gears  —  Variation  of  the  Strength  of  Gear  Teeth  with  the 
Velocity  —  Variation  in  Strength  due  to  Impact  —  The  Effect 

of  Shocks  —  Practical  Considerations  affecting  Design 5I~74 

vii 


yiii  CONTENTS 

CHAPTER  IV 
SIMPLIFIED   FORMULAS  FOR  STRENGTH  OF  GEARS 

PAGES 

Need  for  Simplified  Formulas  —  Assumptions  on  which 
Formulas  are  Based  —  Values  of  Safe  Working  Stress  —  Deri- 
vation of  Simplified  Formulas  —  Correction  for  Numbers  of 
Teeth  —  Correction  for  Increased  Velocity  —  Pinion  Diam- 
eters —  Summary  of  Formulas  —  Chart  for  Rapid  Solution 
of  Gear  Problems  —  Weight  of  Gears  —  Price  of  Gears 75-83 

CHAPTER  V 
THE  STUB-TOOTH  GEAR 

Standards  for  Gear  Teeth  —  Cycloidal  and  Involute  Tooth 
Forms  —  Length  of  Tooth  —  Angle  of  Obliquity,  Efficiency 
and  Wearing  Qualities  —  Investigation  of  an  Actual  Design  — 
Comparison  between  Ordinary  and  Stub  Gear-teeth  —  Fellows 
System  of  Stub  Gear-teeth  —  Nuttall  System  of  Stub  Gear- 
teeth —  Tooth  Shape  for  Rolling  Mill  Gears  —  Strength  of 
Stub  Gear-teeth  —  Advantages  of  Stub-tooth  Systems 84-107 

CHAPTER  VI 
NOISY  GEARING 

Causes  of  Noisy  Gearing  —  Noise  due  to  Interference  — 
Analysis  of  the  Conditions  in  Gears  with  Small  Number  of 
Teeth  —  Charts  for  finding  Number  of  Teeth  in  Large  Gear  — 
Methods  for  avoiding  Interference  —  The  Shortened  Adden- 
dum—  Application  to  Rack  and  Pinion  —  Application  to 
Worm-gearing  —  Application  to  Internal  Gears  —  Application 
to  Bevel  Gears  —  Makeshift  Methods  for  avoiding  Noise  in 
Gearing 108-128 

CHAPTER  VII 
DESIGN  OF  SPUR  GEARING 

Small  and  Medium  Sized  Gears  —  Large  Gears  —  Dimen- 
sions of  Spur  Gears  —  Conditions  governing  the  Design  of 
Gearing  —  Model  Spur  Gear  Drawing  —  Dimensions  of  6ear 
Drawings  —  Rack  Drawing 129-135 


CONTENTS  ix 

CHAPTER  VIII 
METHODS  FOR  CUTTING  SPUR  GEAR  TEETH 

PAGES 

Classification  of  Gear-cutting  Machinery  —  Formed  Tool 
Principle  —  Templet  Principle  —  Odontographic  Principle  — 
Describing-generating  Principle  —  Molding-generating  Prin- 
ciple —  Methods  of  Operation:  By  Impression;  by  Shaping  or 
Planing;  by  Milling;  by  Grinding  or  Abrasion  —  Machines  for 
forming  the  Teeth  of  Spur  Gears  —  Gear  Hobbing  Machines 

—  Designing  a  Hob  for  Hobbing  Spur  Gears  —  Form  and 
Dimensions  of  Tooth  —  Relief  —  Number  of  Flutes  —  Straight 
or  Spiral  Flutes  —  Threading  the  Hob  —  Thread  Relieving 
Tool  —  Heat-treatment     of     Hob  —  Interchangeability     of 
Hobbed  and  Milled  Gears  —  Special  Hobs  for  Gear  Teeth  — 
Generating  Hob  Tooth  Shapes  —  Making  a  Master  Planing 
Tool  for  a  Hob 136-166 

CHAPTER  IX 
PRODUCTION  AND  HEAT-TREATMENT  OF  GEARS 

Processes  in  Production  of  Automobile  Transmission  Gears 

—  Steels  used  for  Automobile  Gears  —  Gear  Blanks  —  Rough- 
turning  —  Annealing  —  Finish-boring  —  Broaching  —  Finish- 
turning  —  Cutting  the  Teeth  —  Rounding  the  Ends  of  the 
Teeth  —  The  Carbonizing  Process  —  Casehardening  and  Oil- 
tempering  —  Sand-blasting  —  Hardness  Testing  —  Grinding  — 
Running-in  —  Heat-treated  Gears  in  Machine  Tools  —  Steel 
to  be  used  —  Advantages  of  Heat-treated  Gears  —  Water  or 
Oil-hardened  Vs.  Casehardened  Gears  —  Points  in  Gear  De- 
sign —  Equipment   for   Heat-treating   Gears  —  Oil   used   for 
Quenching  —  Cost  of  Heat-treated  Gears  —  Kinds  of  Heat- 
treatments  used  —  Tests  on  Heat-treated  Gears  —  Device  for 
Testing  and  Measuring  Gears 167-199 

CHAPTER  X 
BEVEL  GEAR  RULES  AND   FORMULAS 

Bevel  Gear  Definitions  —  Different  Kinds  of  Bevel  Gears  — 
Dimensions  —  Rules  and  Formulas  —  Derivation  of  Formulas 

—  Shafts  at  Right  Angles  —  Miter  Bevel  Gearing  —  Acute- 


X  CONTENTS 

PAGES 

angle  Bevel  Gearing  —  Obtuse-angle  Bevel  Gearing  —  Crown 
Gears  —  Internal  Bevel  Gears  —  Examples  of  Bevel  Gear  Cal- 
culations —  How  to  avoid  Internal  Bevel  Gears  —  Systems  of 
Tooth  Outlines  —  Involute  Teeth  — Octoid  Teeth  —  Milling 
Cutters  for  Bevel  Gears  —  Special  Forms  of  Bevel  Gear 
Teeth 200-223 

CHAPTER  XI 
STRENGTH  AND  DESIGN  OF  BEVEL  GEARS 

Materials  used  for  making  Bevel  Gears  —  Strength  of  Bevel 
Gear  Teeth  —  Rules  and  Formulas  —  Bearing  Pressures  due  to 
the  Action  of  Bevel  Gears  under  Load  —  Example  of  Calcula- 
tion of  Thrust  —  Design  of  Bevel  Gearing  —  Bevel  Gear  Blanks 
—  Model  Bevel  Gear  Drawing  —  Dimensioning  Drawings  for 
Gears  with  Planed  Teeth 224-242 

CHAPTER  XII 
METHODS  FOR  FORMING  THE  TEETH  OF  BEVEL  GEARS 

Comparison  between  Spur  and  Bevel  Gear  Cutting  —  Tred- 
gold's  Approximation  —  Formed  *Tool  Principle  —  Templet 
Principle  —  Odontographic  Principle  —  Molding-generating 
Principle  —  Methods  of  Operation:  By  Impression;  by  Shap- 
ing or  Planing;  by  Milling;  by  Grinding  or  Abrasion  —  Ma- 
chines for  Cutting  the  Teeth  of  Bevel  Gears  by  the  Formed 
Tool  Process  —  Templet  Planing  Machines  —  Adjustable 
Former  for  Bevel  Gear  Planing  —  Bevel  Gear  Generating  Ma- 
chines   243-260 

CHAPTER  XIII 
MILLING  THE  TEETH  IN  BEVEL  GEARS 

Cutting  Bevel  Gears  in  a  Milling  Machine  —  Setting  the 
Cutter  —  Upsetting  and  Rolling  the  Blank  — Trial  Cuts - 
Positive  Determination  of  the  Set-over  —  Filing  the  Teeth  — 
Testing  Bevel  Gear  Teeth  —  Cutting  Bevel  Gears  on  the  Au- 
tomatic Gear-cutting  Machine  —  Derivation  of  Positive  Set- 
over  Formula  —  Practicability  of  Milling  Process  —  Another 
Positive  Method  of  setting  Bevel  Gear  Cutter 261-277 


CONTENTS  Xi 

CHAPTER  XIV 
LONG  AND   SHORT  ADDENDUM   GEARS 

PAGES 

Object  of  Gears  with  Lengthened  Addendum  —  Pinions  with 
Small  Number  of  Teeth  —  Field  of  Application  —  Efficiency  — 
Finding  Circular  Thickness  of  Tooth  —  Duplication  of  Gears 
having  Long  and  Short  Addendum  —  Correction  for  Shrinkage 
in  Hardening 278-290 

CHAPTER  XV 
SKEW  BEVEL  GEARS 

Characteristics  of  Skew  Bevel  Gears  —  Skew  Bevel  Gear 
Tooth  which  can  be  Produced  by  the  Molding-generating  Proc- 
ess—  Method  of  Transmitting  Power  between  Non-parallel, 
Non-intersecting  Shafts  —  Types  of  Skew  Bevel  Gears  —  Theo- 
retical Analysis  of  Proposed  Tooth  Form  —  Practical  Applica- 
tion of  Theory 291-300 


SPUR  AND   BEVEL   GEARING 


CHAPTER  I 
PRINCIPLES  AND  DIMENSIONS  OF  SPUR  GEARING 

GEAR  wheels  are  such  common  objects  in  the  machine  shop, 
and  are  manufactured  with  such  rapidity  and  ease  by  the  aid 
of  the  modern  automatic  gear  cutter,  that  many  seldom  stop  to. 
think  what  they  really  are,  why  the  teeth  must  be  constructed 
with  certain  curves,  and  what  it  is  desired  that  they  shall  accom- 
plish. In  following  chapters  we  shall  take  up  some  of  the 
practical  questions,  touching  upon  the  calculations  that  come 
up  in  the  design,  but  will  here  deal  chiefly  with  a  few  of  the 
theoretical  points  of  the  subject  that  are  seldom  explained  in  a 
simple  manner,  for  the  benefit  of  those  who  have  had  neither 
the  time  nor  the  opportunity  to  look  into  matters  of  this  kind. 

Friction  Wheels.  —  Suppose  there  are  two  wheels  arranged  as 
in  Fig.  i  with  their  faces  in  close,  frictional  contact,  and  that 
both  are  of  exactly  the  same  size,  so  that  when  the  crank  is 
turned  around  once,  wheel  B  will  turn  exactly  once  also,  pro- 
vided, of  course,  there  is  no  slipping  between  the  two  wheels. 
It  must  be  noticed,  moreover,  that  if  the  crank  be  turned  uni- 
formly, wheel  B  will  not  only  make  the  correct  number  of  revolu- 
tions relative  to  A,  but  it  will  revolve  uniformly,  as  well;  that 
is,  both  its  total  motion  and  the  motion  from  point  to  point  will 
be  correct. 

Toothed  Gearing.  —  Now,  there  are  many  places  in  machine 
construction  where  the  slipping  inseparable  from  friction  wheels 
cannot  be  tolerated,  and  this  difficulty  might  be  overcome  by 
fastening  small  projections  to  one  of  the  wheels,  as  on  A  in 
Fig.  2,  and  cutting  grooves  in  the  other  wheel  B.  Then,  if  the 
crank  were  turned,  wheel  B  would  always  make  just  the  right 


number  of  turns,  even  if  considerable  power  were  transmitted. 
It  is  probable,  however,  that  these  projections  and  grooves 
would  not  fulfill  the  purpose  of  gear  teeth.  What  is  wanted  of 
gear  teeth  is  that  they  shall  give  exactly  the  same  kind  of  motion 


Machinery 


Fig.  i.     Two  Wheels  with  Their  Faces  in  Frictional  Contact 

as  corresponding  friction  wheels  running  without  slipping.  They 
must  not  only  keep  the  number  of  revolutions  right,  but  they 
must  give  a  perfectly  even  and  smooth  motion  from  point  to 
point  or  from  tooth  to  tooth. 


Machinery 


Fig.  2.     Example  of  Simplest  Type  of  Toothed  Gearing 

Fig.  3  shows  clearly  how  such  a  result  is  obtained.  It  repre- 
sents the  friction  wheels  with  teeth  fastened  to  them,  the  teeth, 
of  course,  extending  all  the  way  around  instead  of  part  way,  as 
shown.  These  teeth  are  set  so  as  to  be  partly  without  and 


GENERAL  PRINCIPLES  3 

partly  within  the  edges  of  the  two  wheels,  as,  obviously,  they 
will  give  better  results  when  thus  arranged,  than  if  all  the  pro- 
jections were  on  one  wheel,  and  all  the  grooves  or  depressions 
on  the  other,  as  in  Fig.  2. 

With  the  wheels  fitted  in  this  way  it  can  be  proved  that  the 
only  conditions  which  must  be  fulfilled,  in  order  that  the  teeth 
shall  give  to  wheel  B  the  same  motion  that  it  would  have  if  it 
were  driven  by  frictional  contact  with  wheel  A,  is  that  a  line 
drawn  from  the  point  O,  where  the  two  wheels  meet  or  touch 
each  other,  through  the  point  where  the  tooth  curves  touch, 
shall  be  at  right  angles  to  both  tooth  curves  at  this  point,  what- 
ever the  position  of  the  gears.  For  example,  in  Fig.  3,  two  of 


Machinery 


Fig.  3.     Friction  Wheels  with  Properly  Shaped  Gear  Teeth  attached 

to  Them 

the  teeth  touch  at  h.  If  the  curves  are  of  the  right  shape,  a  line 
mn,  drawn  through  h  and  0,  will  be  at  right  angles  to  both 
curves  at  point  h.  This  is  the  law  of  tooth  curves,  and  it  is  of 
no  consequence  what  the  shape  of  the  teeth  is,  so  far  as  their 
correct  action  is  concerned,  if  this  law  holds  true  for  every 
successive  point  where  the  teeth  come  in  contact. 

In  technical  language  the  " friction  wheels"  mentioned  are 
known  as  "pitch  cylinders,"  and  they  are  always  represented  on 
a  gear  drawing  by  a  line  —  usually  a  dash  and  dot  line  —  called 
the  "pitch  line."  As  teeth  are  generally  proportioned,  this  line 
falls  nearly,  but  not  quite,  midway  between  the  tops  and  bottoms 
of  the  teeth,  the  inequality  being  due  to  the  space  left  at  the 


4  SPUR  GEARING 

bottom  of  the  teeth  for  clearance.     The  diameter  of  the  pitch 
cylinder  is  called  the  "pitch  diameter." 

Involute  System.  —  We  are  now  ready  to  consider  the  particu- 
lar forms  of  teeth  most  commonly  used.  The  one  that  is  at 
present  most  in  favor  is  the  involute  tooth,  the  term  "involute" 


Machinery 


Fig.  4.     Method  of  obtaining  Involute  Tooth-curves 

being  the  name  of  a  curve  described  by  the  end  of  a  cord  as  it 
is  unwound  from  another  curve.  For  example,  to  draw  an 
involute,  wind  a  cord  around  a  circular  disk  of  any  convenient 
material,  and  make  a  loop  in  the  outer  end  of  the  cord.  Lay 
the  disk  flat  on  a  piece  of  paper,  and  with  a  pencil-point  passed 
through  the  loop,  unwind  the  string,  keeping  it  drawn  tight, 


GENERAL  PRINCIPLES  5 

and  let  the  point  of  the  pencil  trace  a  curve,  which  will  then  be 
an  involute. 

In  Fig.  4  is  shown  how  this  principle  is  applied  to  forming 
tooth  curves.  A  and  B,  with  centers  at  M  and  N,  are  two 
disks  which  serve  the  purpose  of  pitch  cylinders;  C  and  D  are 
two  smaller  disks  fastened  to  the  larger  ones  and  around  which 
a  cord  is  stretched  and  fastened  at  points  G  and  H.  When 
either  disk  is  turned,  the  cord  is  supposed  to  pull  the  other  one 
around  at  the  same  speed  that  it  would  rotate  if  moved  solely 
by  frictional  contact  between  disks  A  and  B.  To  do  this,  it  is 
simply  necessary  to  have  the  diameters  of  disks  C  and  D  in  the 
same  ratio  as  the  diameters  of  A  and  B.  If  A,  for  example,  is 
half  as  large  as  B,  then  C  must  be  half  as  large  as  D. 

To  make  room  for  drawing  the  curves,  let  pieces  F  and  E  be 
fastened  to  the  large  and  small  wheels,  respectively.  With  a 
pencil  fixed  at  point  d  on  the  cord,  turn  wheel  A  in  the  direction 
of  the  arrow  R,  meanwhile  moving  the  pencil  outward,  and  the 
curve  db  will  be  described,  which  will  be  a  suitable  tooth  curve 
for  the  larger  wheel,  and  which  it  can  be  proved  will  answer  the 
requirements  of  the  general  law.  Starting  again  with  the  pencil 
at  a,  and  turning  wheel  B  in  the  direction  of  the  arrow  P,  and 
moving  the  pencil  outward,  a  similar  curve  ac,  for  the  smaller 
wheel,  will  be  traced. 

The  circles  representing  the  disks  C  and  D  are  called  "base 
circles/7  and  in  practice  are  drawn  at  a  distance  from  the  pitch 
circle  of  about  one-sixtieth  of  the  pitch  diameter.  This  makes 
the  angle  KOd  in  Fig.  4,  called  the  angle  of  obliquity,  about 
14^  degrees;  and  although  it  is  not  by  any  means  certain  that 
this  is  the  best  angle,  it  is  the  one  most  commonly  used. 

Cycloidal  System.  —  Take  a  circular  disk  and  roll  it  along  the 
edge  of  a  ruler  or  straightedge,  holding  the  point  of  a  pencil  at 
the  rim  of  the  disk,  so  that,  as  the  latter  rolls,  the  pencil  will 
trace  a  curve.  This  curve  is  a  cycloid.  Should  the  disk  be 
rolled  on  the  edge  of  another  circular  disk,  however,  the  curve 
traced  would  be  an  epicycloid,  and  should  it  be  rolled  on  the 
inside  of  a  hoop,  it  would  be  called  a  hypocycloid.  These  curves 
are  employed  for  the  teeth  of  the  cycloidal  system  of  gears. 


SPUR  GEARING 


In  Fig.  5  is  shown  how  the  face  or  the  outer  portion  of  the 
tooth  is  rolled  up  by  the  point  A  on  the  outer  rolling  circle,  and 
how  the  flank  or  inner  portion  is  generated  by  point  B  on  the 
inner  rolling  circle.  In  this  case  the  hypocycloid  and  the  flank 
of  the  tooth  are  straight  lines,  the  reason  for  this  being  that,  as 
drawn,  the  diameter  of  the  rolling  circle  is  one-half  the  diameter 
of  the  pitch  circle  of  the  gear,  and  the  hypocycloid  generated 
under  these  conditions  becomes  a  straight  line. 


Machinery 


Fig.  5.     Method  of  obtaining  Cycloidal  Tooth-curves 

Comparison  of  the  Involute  and  Cycloidal  Systems.  —  The 
involute  and  cycloidal  systems  are  the  only  two  that  are  used  to 
any  extent,  and  in  Fig.  6  a  gear  tooth  and  rack  tooth  of  each  are 
shown  for  comparison.  The  involute  gear  tooth  has  the  in- 
volute curve  from  point  a  to  point  b  on  the  base  circle;  from 
b  to  c,  at  the  bottom  of  the  tooth,  the  flank  is  a  straight,  radial 
line.  One  difficulty  with  the  involute  system  is  that,  with  the 
standard  length  of  tooth,  point  a  will  interfere  when  running 
with  gears  or  pinions  having  a  small  number  of  teeth.  To 


GENERAL  PRINCIPLES 


avoid  this,  the  point  is  rounded  off  a  little  below  the  involute 
curve.  In  general  appearance  the  tooth  seems  to  have  a  broad, 
strong  base,  and  a  continuous  curve  from  a  to  c.  A  strong 
feature  of  the  involute  gearing  is  that  it  will  run  correctly  even 
if  the  distance  between  the  centers  of  the  gears  is  not  exactly 
right.  This  will  be  evident  by  referring  to  Fig.  4,  where  it  will 
appear  that  the  relative  velocities  of  the  two  wheels  will  be  the 
same  however  far  apart  they  may  be,  and  if  involute  teeth  are 
used  in  place  of  the  string  connection  there  shown,  the  action  will 
be  just  the  same.  The  involute  rack  tooth  has  straight  sides 
at  an  angle  of  14^  degrees,  with  the  points  slightly  rounded  off. 


Machinery 


Fig.  6.     Comparison  between  Involute  and  Cycloidal  Teeth 

Of  the  cycloidal  teeth  but  little  need  be  said  except  that  they 
have  two  distinct  curves  above  and  below  the  pitch  line,  as 
previously  explained,  and  that  in  the  rack  tooth  the  two  curves 
are  just  alike,  but  reversed. 

Whatever  system  is  used,  it  is  essential  that  all  the  wheels 
of  a  given  pitch  should  be  capable  of  running  together.  To 
make  this  possible  with  the  involute,  all  the  gears  must  have 
the  same  angle  of  obliquity;  and  with  the  cycloidal  system,  the 
same  size  rolling  or  describing  circle  must  be  employed  for  all 


8  SPUR  GEARING 

sizes.  The  circle  generally  chosen  is  one  having  half  the  diam- 
eter of  a  i2-tooth  pinion,  which  makes  the  flanks  of  this  pinion 
radial.  In  Fig.  5,  if  the  diameter  of  the  rolling  circle  had  been 
either  greater  or  less  than  half  the  diameter  of  the  pitch  circle, 
the  flank  of  the  tooth  would  have  been  curved,  and  in  the  case 
of  the  greater  circle,  the  curve  would  have  fallen  inside  of  the 
radial  flank  drawn  in  the  figure,  causing  a  weak,  under-cut  tooth. 
With  the  smaller  circle,  the  curve  would  fall  outside,  making  a 
strong  tooth. 

The  most  important  point  in  favor  of  the  epicycloidal  system 
of  gearing  is  the  freedom  from  interference  of  the  teeth;  but 
this  advantage  is  considerably  modified  by  the  fact  that  it  is 
necessary,  in  order  that  epicycloidal  gears  shall  run  properly 
together,  that  the  pitch  circles  of  the  two  gears  of  a  pair  touch 
or  tangent  each  other,  or,  in  other  words,  that  the  center  distance 
between  the  two  gears  be  exactly  correct.  As  already  men- 
tioned, with  involute  gears  the  distance  between  the  centers 
may  be  varied  somewhat  without  affecting  the  smoothness  with 
which  motion  and  power  may  be  transferred  from  one  gear  to 
another.  The  variation,  however,  must  not  be  great,  on  account 
of  the  fact  that  the  points  of  the  teeth  are  rounded  off  to  avoid 
interference.  The  variation  in  the  center  distance  would,  of 
course,  increase  the  amount  of  backlash,  that  is,  the  space  or 
clearance  between  the  faces  of  the  teeth,  but  the  theoretically 
correct  action  would  not  be  interfered  with.  This  property  of 
involute  gears  is  one  of  the  reasons  why  this  system  has  been 
so  extensively  adopted. 

Of  the  two  systems,  the  epicycloidal  is  the  older.  Cast  gears 
were,  in  the  past,  always  made  with  this  system  of  teeth  and 
many  are  still  so  made,  on  account  of  the  number  of  patterns 
with  this  system  of  teeth  that  are  still  on  hand.  For  cut  gearing, 
however,  and  for  a  large  proportion  of  modern  cast  gearing,  the 
involute  system  has  replaced  the  epicycloidal.  One  objection 
sometimes  brought  forward  against  the  involute  system  is  that 
the  thrust  on  the  shaft  bearings  is  greater  than  when  epicycloidal 
teeth  are  used,  on  account  of  the  obliquity  of  the  line  of  action; 
but  although  it  is  true  that  the  line  of  action  is  at  an  angle  to  the 


GENERAL  PRINCIPLES 


direction  of  the  motion  of  the  gear  teeth,  this  angle  is  a  constant 
angle.  In  the  epicycloidal  system,  on  the  other  hand,  the  line  of 
action  is  at  right  angles  to  a  line  connecting  the  centers  of  the 
two  gears,  when  two  teeth  are  in  contact  on  the  line  of  centers; 
but  the  direction  of  this  pushing  action  is  variable,  so  that  when 
the  teeth  are  coming  in  contact  with  one  another  the  pressure 
has  an  obliquity  fully  as  great  as,  and  sometimes  greater  than, 
that  present  in  standard  involute  gears.  Authorities  on  gearing, 
therefore,  do  not  consider  that  the  objection  mentioned  to  the 
involute  system  of  gear  teeth  has  any  practical  weight. 

Pressure  Angles.  —  While  i^-degree  angle  of  obliquity  or 
pressure  angle  has  been  adopted  as  the  standard  for  involute 
gear  teeth,  it  does  by  no  means  follow  that  all  involute  gear  teeth 


Fig.  7.     Gear-tooth  Parts 

are  made  with  this  angle.  Many  gears  are  made  with  a  20- 
degree  pressure  angle.  This  angle  makes  the  tooth  considerably 
broader  at  the  base  and  correspondingly  narrower  at  the  point. 
The  strength  of  the  tooth  is  thus  increased,  and,  at  the  present 
time,  the  2o-degree  pressure  angle  is  used  to  a  considerable 
extent. 

Definitions  of  Gear-tooth  Parts.  —  When  one  of  two  gears 
that  are  in  mesh  with  each  other  is  revolved,  it  will  drive  the 
other  gear  at  a  certain  rate.  Imagine,  as  has  already  been  ex- 
plained, that  instead  of  gears  two  circular  disks  are  in  contact, 
so  that  when  one  disk  is  revolved  it  will  drive  the  other  disk  by 
frictional  force.  The  diameters  of  the  disks  may  be  so  selected 
that  when  one  revolves  at  the  same  rate  as  the  gear  to  which  it 
corresponds,  it  will  drive  the  other  disk  at  the  same  rate  as  the 


10  SPUR  GEARING 

second  gear  would  be  driven.  The  diameters  of  the  disks  are 
then  the  same  as  the  pitch  diameters  of  the  gears,  and  the  circum- 
ferences of  these  disks,  which  are  tangent  or  touch  each  other, 
represent  the  pitch  circles  of  the  gears. 

The  outside  diameter  of  a  gear  is  the  diameter  measured  over 
the  tops  of  the  teeth. 

The  root  diameter  of  a  gear  is  the  diameter  measured  at  the 
bottom  or  roots  of  the  teeth. 

The  center  distance  is  the  distance  between  the  centers  of  two 
meshing  gears,  the  pitch  circles  of  which  are  tangent  to  each 
other. 

The  diametral  pitch  of  a  gear  is  the  number  of  teeth  for  each 
inch  of  pitch  diameter,  and  is  found  by  dividing  the  number  of 
teeth  by  the  pitch  diameter. 

The  circular  pitch  is  the  distance  from  the  center  of  one  tooth 
to  the  center  of  the  next  along  the  pitch  circle. 

The  chordal  pitch  is  the  distance  from  the  center  (on  the  pitch 
circle)  of  one  tooth  to  the  center  of  the  next,  measured  along  a 
straight  line. 

The  thickness  of  the  tooth  is  generally  understood  to  be  the 
thickness  at  the  pitch  circle,  measured  along  the  circular  arc. 

The  chordal  thickness  of  the  tooth  is  the  thickness  at  the  pitch 
circle  measured  along  a  straight  line  or  chord. 

The  addendum  of  a  gear  tooth  is  the  distance  from  the  pitch 
circle  to  the  top  of  the  tooth. 

The  dedendum  of  a  gear  tooth  is  the  distance  from  the  pitch 
circle  to  the  root  of  the  tooth. 

The  working  depth  is  the  depth  to  which  the  teeth  in  a  meshing 
gear  enter  into  the  spaces  between  the  teeth. 

The  clearance  is  the  amount  by  which  the  tooth  space  is  cut 
deeper  than  the  working  depth. 

The  face  of  the  tooth  is  that  part  of  the  tooth  curve  that  is 
between  the  outside  circumference  and  the  pitch  circle. 

The  flank  of  the  tooth  is  that  part  of  the  working  depth  of  the 
tooth  which  comes  inside  of  the  pitch  circle. 

In  the  following  will  be  given  a  number  of  rules  and  examples 
showing  the  relation  between  these  various  dimensions. 


GENERAL  PRINCIPLES 


II 


3  P 


4  P 


5  P 


6  P 


7  P 


8  P 


Fig.  8.  Gear  Teeth  of  Different  Diametral  Pitch  shown  in  Full  Natural 
Size  in  order  to  enable  the  Designer  to  Determine  at  a  Glance  the 
Actual  Proportions  of  Gear  Teeth  of  Various  Pitches 


12  SPUR  GEARING 

Circular  and  Diametral  Pitch.  —  The  circular  pitch,  at  the 
present  time,  is,  as  a  rule,  used  only  in  relation  to  gears  with  cast 
teeth,  which  are  not  afterwards  finished  or  cut.  Diametral  pitch 
is  used  almost  exclusively  for  all  cut  gearing,  and  of  late,  to  some 
extent,  for  cast  gearing  as  well. 

As  the  circular  pitch  is  equal  to  the  distance  from  the  center  of 
one  tooth  to  the  center  of  the  next,  measured  along  the  pitch 
circle,  it  can  readily  be  seen  that  the  circular  pitch  will  be  equal 
to  the  circumference  of  the  pitch  circle  divided  by  the  number  of 
teeth  in  the  gear.  The  circumference,  however,  is  equal  to  the 
pitch  diameter  multiplied  by  3.1416;  hence,  we  can  write  the  rule 
for  finding  the  circular  pitch,  when  the  pitch  diameter  of  the  gear 
and  the  number  of  teeth  in  the  gear  are  known,  as  follows: 

^.      !        ..  ,       pitch  diameter  X  3.1416 
Circular  pitch  =  K  ^ 

number  of  teeth 

Example:  —  The  pitch  diameter  of  a  gear  is  47!  inches  and  the 
gear  has  75  teeth.  Find  the  circular  pitch. 

The  circumference  of  the  pitch  circle  is  47!  X  3.1416  =  150.01 
inches;  then  the  circular  pitch  equals  150.01  -f-  75  =  2  inches, 
almost  exactly. 

The  relation  between  the  diametral  pitch,  the  pitch  diameter 
and  the  number  of  teeth  is  much  simpler  than  that  between  the 
circular  pitch  and  these  quantities.  As  the  diametral  pitch  is 
the  number  of  teeth  for  each  inx:h  of  pitch  diameter,  it  can  readily 
be  seen  that  the  diametral  pitch  can  be  found  by  dividing  the 
number  of  teeth  by  the  pitch  diameter. 

If  a  gear  has  20  teeth  and  the  pitch  diameter  is  2  inches,  the 
diametral  pitch  would  be  equal  to  20  -f-  2,  or  10.  In  the  same 
way,  if  the  gear  has  22  teeth  and  the  pitch  diameter  is  5!  inches, 
then  the  diametral  pitch  equals  22  -f-  5^  =  4.  The  relationship 
between  diametral  pitch,  pitch  diameter  and  the  number  of  teeth 
may  be  written  as  below: 

TV  ,    .,  ,       number  of  teeth 

Diametral  pitch  =  — ; — : — 

pitch  diameter 

Changing  Circular  Pitch  into  Diametral  Pitch.  —  If  the 
circular  pitch  is  given  and  the  diametral  pitch  is  to  be  found, 


DIAMETRAL  AND   CIRCULAR  PITCH  13 

this  can  be   done  by  dividing  3.1416  by  the  circular  pitch, 
or: 

Diametral  pitch  =    .    3;I4I<S.    ^ 
circular  pitch 

Example:  —  If  the  circular  pitch  of  a  gear  is  f  inch,  what  is 
the  nearest  whole  number  diametral  pitch? 

By  following  the  rule  given  above,  we  have  3. 14 1 6  -f-  f  =  5.026, 
which  shows  that  5  is  the  nearest  whole  number  diametral  pitch 
corresponding  to  a  f -inch  circular  pitch. 

Changing  Diametral  Pitch  into  Circular  Pitch.  —  If  the 
diametral  pitch  is  known  and  the  circular  pitch  is  to  be  found, 
divide  3.1416  by  the  diametral  pitch.  The  quotient  is  the  circu- 
lar pitch,  or: 

3-1416 


Circular  pitch  = 


diametral  pitch 


Example:  —  If  the  diametral  pitch  of  a  gear  is  4,  what  is  the 
corresponding  circular  pitch? 

According  to  the  rule  given,  3.1416  -r-  4  =  0.7854  inch  is  the 
circular  pitch  of  the  gear. 

Explanation  of  Rules  Given.  —  Having  given  the  rules,  we 
will  now  proceed  to  explain  how  they  are  obtained.  We  know 
that  the  distance  around  the  circumference  of  a  circle  is  equal  to 
3.1416  multiplied  by  the  diameter  of  the  circle;  hence,  for  every 
inch  of  diameter,  we  have  3.1416  inches  of  circumference.  As 
the  diametral  pitch  of  a  gear  is  equal  to  the  number  of  teeth  for 
each  inch  of  pitch  diameter,  and  each  inch  of  diameter  is  repre- 
sented by  3.1416  inches  of  circumference,  then  the  diametral 
pitch  equals  the  number  of  teeth  for  each  3.1416  inches  of  cir- 
cumference. As  the  circular  pitch  is  the  distance  from  the 
center  of  one  tooth  to  the  center  of  the  next,  then  the  circular 
pitch  must  be  equal  to  3.1416  divided  by  the  number  of  teeth 
in  that  3.1416  inches  of  circumference,  and,  as  we  have  shown 
that  the  diametral  pitch  is  equal  to  the  number  of  teeth  in  each 
3.1416  inches  of  circumference,  then  the  circular  pitch  must  equal 
3.1416  divided  by  the  diametral  pitch  as  given  by  the  rule  and 
formula. 


14  SPUR  GEARING 

It  may  not  be  actually  necessary  to  show  how  we  obtain  the 
diametral  pitch  from  the  circular  pitch,  because  the  formula  may 
be  simply  transposed  to  give  the  required  result.  For  the  sake 
of  completeness,  however,  the  explanation  will  be  given.  As  in 
the  preceding  case,  we  begin  with  the  ratio  of  the  circumference 
of  the  circle  to  its  diameter,  which  is  3.1416.  In  each  3.1416 
inches  of  circumference  we  have  a  certain  number  of  teeth,  which 
is  the  diametral  pitch  of  the  gear.  Now,  having  given  the  circu- 
lar pitch,  if  we  divide  3.1416  by  that,  we  obtain  the  number 
of  teeth  for  3.1416  inches  of  the  circumference,  which  is  the 
diametral  pitch  of  the  gear,  as  shown  by  the  rule  and  formula 
given. 

Tables  of  gear  tooth  parts  are  given  in  the  following  pages 
which  will  facilitate  the  finding  of  corresponding  diametral  and 
circular  pitches.  These  tables  give,  in  addition,  other  dimen- 
sions relating  to  gear  teeth,  such  as  the  thickness  of  the  tooth 
at  the  pitch  line,  the  addendum,  the  working  depth  of  the  tooth, 
the  depth  of  space  below  the  pitch  line  (or  dedendum),  and  the 
whole  depth  of  the  tooth. 

Pitch  Diameter.  —  When  the  diametral  pitch  and  the  number 
of  teeth  in  a  gear  are  known,  the  pitch  diameter  is  found  by 
dividing  the  number  of  teeth  by  the  diametral  pitch,  or: 

TV,  ,    ,.  number  of  teeth 

Pitch  diameter  =  — - 

diametral  pitch 

Example:  —  A  lo-pitch  gear  has  35  teeth.  Find  the  pitch 
diameter. 

Divide  35  (the  number  of  teeth)  by  10  (the  diametral  pitch). 
The  quotient  is  3^,  which  is  the  pitch  diameter  of  the  gear  in 
inches. 

That  the  rule  and  formula  given  above  is  correct,  is  proved  by 
the  definition  of  diametral  pitch.  As  the  diametral  pitch  equals 
the  number  of  teeth  for  each  inch  of  pitch  diameter,  then,  if  we 
divide  the  number  of  teeth  in  the  gear  by  the  diametral  pitch, 
we  must  obviously  obtain  the  number  of  inches  of  the  pitch 
diameter. 

When  the  circular  pitch  and  the  number  of  teeth  in  a  gear 


DIMENSIONS  15 

are  given,  the  pitch  diameter  is  found  by  dividing  the  product 
of  the  number  of  teeth  and  the  circular  pitch  by  3.1416,  or: 

„..  ,,.  number  of  teeth  X  circular  pitch 

Pitch  diameter  = *f — 

3.1416 

Example:  —  A  gear  of  2-inch  circular  pitch  has  75   teeth. 
Find  the  pitch  diameter. 
According  to  the  rule  given,  the  pitch  diameter  equals: 

75  X  2  .     ,  , 

** ~  =  47.75  inches,  very  nearly. 

3.1416 

The  accuracy  of  the  rule  and  formula  given  for  finding  the 
pitch  diameter  when  the  circular  pitch  is  given  will  readily  be 
understood.  As  the  circular  pitch  is  the  distance  from  the 
center  of  one  tooth  to  the  next,  this  distance  multiplied  by  the 
total  number  of  teeth  will  give  the  total  pitch  circumference  of 
the  gear.  This  divided  by  3.1416,  of  course,  gives  the  pitch 
diameter. 

Center  Distance.  —  To  find  the  center  distance  when  the 
numbers  of  teeth  in  the  two  gears  and  the  diametral  pitch  are 
given,  add  together  the  number  of  teeth  in  both  gears  and  divide 
the  sum  by  two  times  the  diametral  pitch,  or: 

Center  distance  =  no-  of  teeth>  ist  £ear  +  no.  of  teeth,  2nd  gear 

2  X  diametral  pitch 

Example:  —  Find  the  center  distance  between  two  gears  of 
8  diametral  pitch,  the  number  of  teeth  in  the  one  gear  being  22 
and  in  the  other,  36. 

6  58  a     . 

^  =  3f  mches. 

This  formula  is  based  upon  the  fact  that  the  pitch  diameter  of 
each  of  the  gears  equals  the  number  of  teeth  in  that  gear,  divided 
by  the  diametral  pitch.  The  sum  of  the  teeth  in  the  two  gears, 
divided  by  the  diametral  pitch,  will  then  equal  the  sum  of  the 
two  pitch  diameters,  but  as  the  center  distance  is  equal  to  the  sum 
of  the  pitch  radii,  and  is  thus  equal  to  only  one-half  of  the 
total  sum  of  the  two  pitch  diameters,  this  sum  is  divided  by  2, 


16  SPUR  GEARING 

as  indicated  in  the  rule  and  formula,  to  obtain  the  distance 
between  the  centers. 

To  find  the  center  distance  when  the  circular  pitch  and  the 
numbers  of  teeth  in  the  two  gears  are  given,  multiply  the  suiri 
of  the  number  of  teeth  in  both  gears  by  the  circular  pitch  and 
divide  the  product  by  6.2832,  or: 

Center  distance  = 

(no.  of  teeth,  ist  gear  +  no.  of  teeth,  2nd  gear)  X  circ.  pitch 

6.2832 

Example:  —  The  circular  pitch  of  two  equal  gears  having  75 
teeth  each  is  2  inches.  Find  the  center  distance. 

(75  +  75)   X  2  =  ^^^^  ^^ 

6.2832 

This  rule  is  derived  directly  from  the  one  just  given  in  which 
the  diametral  pitch  is  used,  by  merely  substituting  the  circular 
pitch  for  the  diametral  pitch,  as  explained  on  a  preceding  page. 

Addendum. — The  addendum  of  a  gear  tooth  is  always,  in 
standard  diametral  pitch  gearing,  made  equal  to  i  divided  by 
the  diametral  pitch,  or: 


Addendum  = 


diametral  pitch 

Example:  —  Find  the  addendum  for  a  gear  of  6  diametral 
pitch.  The  addendum  equals  i  -s-  6,  or  0.1667  inch. 

If  the  circular  pitch  is  given,  the  addendum  equals  the  circular 
pitch  divided  by  3.1416,  or: 

AJJ     j  circular  pitch 

Addendum  =  •= 

3.1416 

Example:  —  The  circular  pitch  is  2  inches.  Find  the 
addendum. 

According  to  the  rule  given,  the  addendum  equals  2  -f-  3.1416 
=  0.6366. 

This  rule  is  derived  directly  from  that  for  diametral  pitch,  by 
substituting  the  value  of  the  circular  pitch  for  the  diametral 
pitch  in  that  rule. 


DIMENSIONS  17 

Clearance.  —  The  clearance  below  the  working  depth  of  the 
tooth  is  made  equal  to  0.157  divided  by  the  diametral  pitch,  or; 

Clearance  =  — — 0mI$J  .    . 
diametral  pitch 

Example:  —  Find  the  clearance  for  1 2  diametral  pitch. 

Clearance  =       ^7  =  0.013  inch. 
12 

If  the  circular  pitch  is  given,  the  clearance  may  be  found  by 
dividing  the  circular  pitch  by  20,  or: 

^  circular  pitch 

Clearance  = c 

20 

Example:  —  Find  the  clearance  for  a  gear  of  2-inch  circular 
pitch. 

Clearance  =  fa  =  o.i  inch. 

Clearance  of  Gears  cut  on  the  Gear  Shaper.  —  When  gears 
are  cut  on  the  Fellows  gear  shaper,  the  clearance  is  made  equal  to 
0.25  divided  by  the  diametral  pitch;  hence,  the  root  diameter  of 
these  gears  is  smaller  than  the  root  diameter  of  ordinary  milled 
gears.  The  pitch  and  outside  diameters  are,  of  course,  the  same 
as  for  gears  milled  with  ordinary  rotary  milling  cutters. 

Whole  Depth  of  Tooth.  —  The  whole  depth  of  tooth  is  com- 
posed of  two  times  the  addendum  plus  the  clearance,  and,  there- 
fore, is  found  by  dividing  2.157  by  the  diametral  pitch,  or: 

Whole  depth  = 2'1517  .    T 

diametral  pitch 

Example: — Find  the  whole  depth  of  tooth  for  5  diametral  pitch. 

/y     T  C  *7 

Whole  depth  =       0/-  =  0.431  inch, 
o 

If  the  circular  pitch  is  given,  the  whole  depth  of  the  tooth  is 
found  by  multiplying  the  circular  pitch  by  0.6866,  or: 

Whole  depth  =  circular  pitch  X  0.6866. 

Example:  —  Find  the  whole  depth  of  tooth  for  2-inch  circular 
pitch. 

Whole  depth  =  2  X  0.6866  =  1.3732  inch. 


18  SPUR  GEARING 

The  rule  for  finding  the  whole  depth  of  tooth  when  the  circular 
pitch  is  given  may  be  found  from  the  rule  relating  to  diametral 
pitch,  by  simply  substituting  the  value  of  the  circular  pitch  for 
that  of  the  diametral  pitch. 

Thickness  of  Tooth.  —  The  thickness  of  the  tooth  at  the  pitch 
line,  measured  along  the  circular  arc,  is  found  by  dividing  1.5708 
by  the  diametral  pitch,  or: 

1.5708 


Thickness  of  tooth 


diametral  pitch 


Example:  —  Find  the  thickness  of  the  tooth  at  the  pitch  line 
for  a  gear  of  4  diametral  pitch. 

i  ^708 

Thickness  of  tooth  =  -1^-L  -  =  0.3927  inch. 

4 

This  rule  makes  the  tooth  space  and  the  thickness  of  the  tooth 
at  the  pitch  line  exactly  equal. 

If  the  circular  pitch  is  known,  the  thickness  of  the  tooth  equals 
the  circular  pitch  divided  by  2,  or: 

r^,  .  ,  r  ,      .,       circular  pitch 

Thickness  of  tooth  =  - 

2 

Example:  —  Find  the  thickness  of  tooth  for  a  if -inch  circular 
pitch  gear. 

Thickness  of  tooth  =  —  =  J  inch. 

2 

It  is  apparent  at  a  glance  that  this  rule  is  based  upon  the  rule 
of  making  the  thickness  of  the  tooth  equal  to  the  tooth  space, 
because  the  thickness  of  one  tooth  plus  one  tooth  space  equals 
the  circular  pitch,  and  the  thickness  of  the  tooth  is  then  made 
one-half  of  this  distance. 

Outside  Diameter.  —  When  the  diametral  pitch  is  known,  the 
outside  diameter  of  a  gear  is  found  by  adding  2  to  the  number  of 
teeth  and  dividing  the  sum  thus  obtained  by  the  diametral  pitch, 
or: 

number  of  teeth  +  2 


Outside  diam.  = 


diametral  pitch 


DIMENSIONS  19 

Example:  —  Find  the  outside  diameter  of  a  5  diametral  pitch 
gear  having  28  teeth. 

Outside  diam.  =  -  -  -  =  ^  =  6  inches. 
5  5 

If  the  circular  pitch  is  given,  the  outside  diameter  is  found  by 
multiplying  the  sum  of  the  number  of  teeth  plus  2  by  the  circular 
pitch  and  dividing  the  product  by  3.1416,  or: 

~  ,  .j     ,.  (no.  of  teeth  +  2)  X  circ.  pitch 

Outside  diam.  =  »  -  !  —  «  -  ^  - 

3.1416 

Example:  —  Find  the  outside  diameter  of  a  2-inch  circular 
pitch  gear  having  75  teeth. 

(75  +  2)  X  2      77  X  2         154 
****-*  —  '—  -  =  "  -  -  =  —  ^—  =  40.02  uiches. 
3.1416  3.i4J6       3.i4J6 

Number  of  Teeth.  —  If  the  diametral  pitch  and  pitch  diameter 
are  known,  multiply  the  pitch  diameter  by  the  diametral  pitch. 
The  product  gives  the  number  of  teeth  in  the  gear,  or: 

No.  of  teeth  =  pitch  diameter  X  diametral  pitch. 

Example:  —  Find  the  number  of  teeth  of  an  8  diametral  pitch 
gear  having  a  pitch  diameter  of  3!  inches. 

Number  of  teeth  =  3!  X  8  =  27. 

When  the  circular  pitch  and  the  pitch  diameter  are  known,  the 
number  of  teeth  are  found  by  multiplying  the  pitch  diameter  by 
3.1416  and  dividing  the  product  by  the  circular  pitch,  or: 


No.  of  teeth  =  3-^i6  X  pitch  diameter 
circular  pitch 

Example:  —  Find  the  number  of  teeth  in  a  2-inch  circular 
pitch  gear  having  a  pitch  diameter  of  47!  inches. 

'       '     No.  of  teeth  = 


Miscellaneous  Rules.  —  If  the  outside  diameter  is  known  and 
the  pitch  diameter  is  to  be  found,  subtract  2  times  the  addendum 
from  the  outside  diameter,  or: 

Pitch  diam.  =  outside  diam.  —  2  X  addendum. 


20  SPUR  GEARING 

Example:  —  Find  the  pitch  diameter  of  an  8  diametral  pitch 
gear  having  an  outside  diameter  of  4  inches. 

As  the  addendum  equals  i  divided  by  the  diametral  pitch,  we 
have,  in  this  case: 

Addendum  =  J 

Pitch  diameter  =  4  —  2  X  f  =  3!  inches. 

If  the  outside  diameter  and  the  number  of  teeth  are  known,  the 
pitch  diameter  may  be  found  by  multiplying  the  outside  diameter 
by  the  number  of  teeth  and  dividing  this  product  by  the  number 
of  teeth  plus  2,  or: 

TV  *  ^  j-  outside  diam.  X  no.  of  teeth 

Pitch  diam.  =  - 

no.  of  teeth  +  2 

Example:  —  A  gear  of  3^  inches  outside  diameter  has  36  teeth. 
Find  the  pitch  diameter. 
According  to  the  rule  given: 

Pitch  diameter  =  3**3g6  =  ~  =  3  inches. 

If  the  outside  diameter  and  the  diametral  pitch  are  known,  the 
number  of  teeth  may  be  found  by  multiplying  the  outside  diam- 
eter by  the  diametral  pitch  and  subtracting  2  from  the  product, 
or: 

No.  of  teeth  =  (outside  diam.  X  diametral  pitch)  —  2. 

Example:  —  Find  the  number  of  teeth  in  a  gear  of  3  £  inches 
outside  diameter  and  12  diametral  pitch. 
According  to  the  rule  given: 

Number  of  teeth  =  (3^  X  12)  —  2  =  36. 

Table  of  Rules  and  Formulas.  —  For  those  who  prefer  rules 
and  formulas  in  a  condensed,  form,  the  accompanying  table 
entitled  "Rules  and  Formulas  for  Dimensions  of  Spur  Gears" 
has  been  prepared.  By  grouping  the  formulas  and  dimensions 
together  in  this  manner,  they  may  be  more  easily  found  when 
wanted.  The  formulas  are  numbered,  making  reference  to  any 
of  them  more  convenient.  The  first  sixteen  formulas  are  placed 
in  the  order  in  which  they  would  naturally  present  themselves 
to  the  designer  when  determining  the  dimensions  of  a  pair  of 


DIMENSIONS 


21 


Rules  and  Formulas  for  Dimensions  of  Spur  Gears  * 


No.  of 
Rule 

To  Find 

Rule 

Formula 

i 

Diametral 
Pitch 

Divide  3.1416  by  circular  pitch. 

P      3.I4I6 
P' 

2 

Circular 
Pitch 

Divide  3.1416  by  diametral  pitch. 

P,  ,  3.i4i6 
P 

3 

Pitch 
Diameter 

Divide  number  of  teeth  by  diametral  pitch. 

M 

4 

Pitch 
Diameter 

Multiply  number  of  teeth  by  circular  pitch  and 
divide  the  product  by  3.1416. 

£>--££ 

3-1416 

5 

Center 
Distance 

Add  the  number  of  teeth  in  both  gears  and 
divide  the  sum  by  two  times  the  diametral 
pitch. 

_      Ng  +  NP 

C=         2P 

Center 

Multiply  the  sum  of  the  number  of  teeth  in 

c^(Ng  +  NP)P> 

Distance  ' 

product  by  6.2832. 

6.2832 

7 

Addendum 

Divide  I  by  diametral  pitch. 

H  ? 

8 

Addendum 

Divide  circular  pitch  by  3.1416. 

3-1416 

9 

Clearance 

Divide  0.157  by  diametral  pitch. 

F  =  o.lS7 
P 

10 

Clearance 

Divide  circular  pitch  by  20. 

f.e 

20 

ii 

Whole  Depth 
of  Tooth 

Divide  2.157  by  diametral  pitch. 

ur     2'157 
W~     P 

12 

Whole  Depth 
of  Tooth 

Multiply  0.6866  by  circular  pitch. 

W  =  0.6866  P' 

13 

Thickness 
of  Tooth 

Divide  1.5708  by  diametral  pitch. 

T  _  I-S7Q8 
P 

14 

Thickness 
of  Tooth 

Divide  circular  pitch  by  2. 

r=£' 

2 

IS 

Outside 
Diameter 

Add  2  to  the  number  of  teeth  and  divide  the 
sum  by  diametral  pitch. 

o  =  ^±-2 
P 

16 

Outside 
Diameter 

Multiply  the  sum  of  the  number  of  teeth  plus 
2  by  circular  pitch  and  divide  the  product  by 
3-1416. 

0==    (AT  +  2)P' 
3-1416 

17 

Diametral 
Pitch 

Divide  number  of  teeth  by  pitch  diameter. 

-| 

18 

Circular 

Pitch 

Multiply  pitch  diameter  by  3.1416  and  divide 
by  number  of  teeth. 

r      3.1416  D 

N 

19 

Pitch 
Diameter 

Subtract  two  times  the  addendum  from  out- 
side diameter. 

D  =  O-zS 

20 

Number  of 
Teeth 

Multiply  pitch  diameter  by  diametral  pitch. 

N  =  PXD 

21 

Number  of 
Teeth 

Multiply  pitch  diameter  by  3.1416  and  divide 
the  product  by  circular  pitch. 

N  _  3.1416  D 
P' 

22 

Outside 
Diameter 

Add  two  times  the  addendum  to  the  pitch 
diameter. 

O  =  D  +  2S 

23 

Length  of 
Rack 

Multiply  number  of  teeth  in  rack  by  3.1416  and 
divide  by  diametral  pitch. 

L      3.I4I6  N 
P 

24 

Length  of 
Rack 

Multiply  the  number  of  teeth  in  the  rack  by 
circular  pitch. 

L  =  NP' 

From  MACHINERY'S  HANDBOOK,  page  550. 


22 


SPUR  GEARING 


Gear  Tooth  Parts  * 

(Diametral  Pitch  Gears) 


Diam- 
etral 
Pitch 

Circular 
Pitch 

Thickness 
of  Tooth  on 
Pitch  Line 

Addendum 

Working 
Depth  of 
Tooth 

Depth  of 
Space  below 
Pitch  Line 

Whole 
Depth  of 
Tooth 

P 

P' 

T 

S 

W 

S  +  F 

W 

H 

6.2832 

3.1416 

2.OOOO 

4.0000 

2.3142 

4.3H2 

H 

4.1888 

2.0944 

1-3333 

2.6666 

1.5428 

2.8761 

i 

3.1416 

1.5708 

i  .0000 

2.0000 

I.I57I 

2.1571 

iH 

2.5133 

1.2566 

0.8000 

I.  "6000 

0.9257 

I.7257 

iH 

2.0944 

1.0472 

0.6666 

1-3333 

0.7714 

1.4381 

1% 

•7952 

0.8976 

o.57i4 

1.1429 

0.6612 

1.2326 

2 

.5708 

0.7854 

0.5000 

i  .0000 

0.5785 

1.0785 

2}4 

.3963 

0.6981 

0.4444 

0.8888 

o.5i43 

0.9587 

2H 

.2566 

0.6283 

0.4000 

0.8000 

0.4628 

0.8628 

2H 

.1424 

0.5712 

0.3636 

0.7273 

0.4208 

0.7844 

3 

.0472 

0.5236 

0-3333 

0.6666 

0.3857 

0.7190 

3U 

0.8976 

0.4488 

0.2857 

o.57i4 

0.3306 

0.6163 

4 

0.7854 

0.3927 

0.2500 

0.5000 

0.2893 

0-5393 

5 

0.6283 

0.3142 

O.2OOO 

0.4000 

0.2314 

0.43*4 

6 

0.5236 

0.26l8 

0.1666 

0-3333 

0.1928 

0-3595 

7 

0.4488 

0.2244 

0.1429 

0.2857 

0.1653 

0.3081 

•     8 

0.3927 

0.1963 

0.1250 

0.2500 

0.1446 

0.2696 

9 

0.3491 

0.1745 

O.IIII 

0.2222 

0.1286 

0.2397 

10 

0.3142 

0.1571 

0.1000 

0.2000 

0.1157 

0.2157 

ii 

0.2856 

0.1428 

0.0909 

0.1818 

0.1052 

0.1961 

12 

0.2618 

0.1309 

0.0833 

0.1666 

0.0964 

0.1798 

13 

0.2417 

o.i  208 

0.0769 

0.1538 

0.0890 

0.1659 

14 

0.2244 

O.II22 

0.0714 

0.1429 

0.0826 

0.1541 

15 

o  .  2094 

0.1047 

0.0666 

0.1333 

0.0771 

0.1438 

16 

0.1963 

0.0982 

0.0625 

0.1250 

0.0723 

0.1348 

i? 

0.1848 

0.0924 

0.0588 

o  .  1  1  76 

0.0681 

0.1269 

18 

0.1745 

0.0873 

0.0555 

O.IIII 

0.0643 

0.1198 

iQ 

0.1653 

0.0827 

0.0526 

0.1053 

0.0609 

O.H35 

20 

0.1571 

0.0785 

0.0500 

0.1000 

0.0579 

0.1079 

22 

0.1428 

0.0714 

0.0455 

0.0909 

0.0526 

o  .  0980 

24 

0.1309 

0.0654 

0.0417 

0.0833 

0.0482 

0.0898 

26 

0.1208 

o  .  0604 

0.0385 

0.0769 

0.0445 

0.0829 

28 

O.II22 

0.0561 

0.0357 

0.0714 

0.0413 

0.0770 

30 

o  .  1047 

0.0524 

0.0333 

0.0666 

0.0386 

0.0719 

32 

0.0982 

O.O49I 

0.0312 

0.0625 

0.0362 

0.0674 

34 

0.0924 

0.0462 

0.0294 

0.0588 

0.0340 

0.0634 

36 

0.0873 

0.0436 

0.0278 

0.0555 

0.0321 

0.0599 

38 

0.0827 

0.0413 

0.0263 

0.0526 

0.0304 

0.0568 

40 

0.0785 

0.0393 

0.0250 

0.0500 

0.0289 

0.0539 

42 

0.0748 

0.0374 

0.0238 

0.0476 

0.0275 

0.0514 

44 

0.0714 

0.0357 

0.0227 

0.0455 

0.0263 

o  .  0490 

46 

0.0683 

0.0341 

0.0217 

0.0435 

0.0252 

o  .  0469 

48 

0.0654 

0.0327 

0.0208 

0.0417 

0.0241 

o  .  0449 

So 

0.0628 

0.0314 

O.O2OO 

0.0400 

0.0231 

0.0431 

*  From  MACHINERY'S  HANDBOOK,  page  552. 


DIMENSIONS 


Gear  Tooth  Parts  * 

(Circular  Pitch  Gears) 


Circular 

Diametral 

Thickness 
of  Tooth  on 

Addendum 

Working 
Depth  of 

Depth  of 
Space  below 

Whole 
Depth  of 

Pitch 

Pitch 

Pitch  Line 

Tooth 

Pitch  Line 

Tooth 

P' 

P 

r 

* 

W 

S  +  F 

W 

4 

0.7854 

2.OOOO 

1.2732 

2  •  5464 

1-4732 

2.7464 

3^5 

0.8976 

.7500 

I  .1140 

2.2281 

I  .  2890 

2.4031 

3g 

.0472 

.5OOO 

0-9549 

1.9098 

I  .  1049 

2.0598 

.1424 

•3750 

0-8753 

.7506 

1.0128 

.8881 

2H 

.2566 

.25OO 

0-7957 

•5915 

0.9207 

.7165 

2H 

•3963 

.I25O 

0.7162 

•4323 

0.8287 

.5448 

2 

.5708 

.OOOO 

0.6366 

.2732 

0.7366 

•3732 

1% 

•6755 

0-9375 

0.5968 

•1937 

o  .  6906 

.2874 

1% 

•7952 

0.8750 

0.5570 

.1141 

0.6445 

.2016 

i$i 

•9333 

0.8125 

0.5173 

•0345 

0.5985 

.1158 

iH 

2.0944 

0.7500 

0-4775 

0-9549 

0.5525 

.0299 

i7Ao 

2.1855 

0.7187 

0.4576 

0.9151 

0-5294 

0.9870 

iH 

2  .  2848 

0.6875 

0-4377 

0.8754 

0.5064 

0.9441 

IMe 

2.3936 

0.6562 

0.4178 

0.8356 

0.4834 

0.9012 

iH 

2.5133 

0.6250 

0-3979 

0.7958 

o  .  4604 

0.8583 

IMe 

2.6456 

0-5937 

0.3780 

0.7560 

0.4374 

0.8154 

1^6 

2.7925 

0.5625 

0.3581 

0.7162 

0.4143 

0.7724 

I  He 

2.9568 

0.5312 

0.3382 

0.6764 

0.3913 

0.7295 

I 

3.I4I6 

0.5000 

0.3183 

0.6366 

0.3683 

0.6866 

iMe 

3.3510 

0.4687 

0.2984 

0.5968 

0.3453 

0.6437 

% 

3.5904 

0-4375 

0.2785 

0.5570 

0.3223 

0.6007 

iMe 

3.8666 

0.4062 

0.2586 

0.5173 

0.2993 

0-5579 

% 

4.1888 

o.375o 

0.2387 

0-4775 

0.2762 

0-5150 

'Me 

4.5696 

0-3437 

0.2189 

0-4377 

0.2532 

0.4720 

% 

4.7124 

0-3333 

O.  2122 

0.4244 

0.2455 

0-4577 

N 

5.0265 

0.3125 

0.1989 

0-3979 

0.2301 

0.4291 

Me 

5-5851 

0.2812 

0.1790 

0.3581 

0.2071 

0.3862 

H 

6.2832 

0.2500 

0.1592 

0.3183 

0.1842 

0-3433 

Me 

7  .  1808 

0.2187 

0.1393 

0.2785 

0.1611 

0.3003 

% 

7  •  8540 

O.2OOO 

0.1273 

0.2546 

0.1473 

0.2746 

H 

8.3776 

0.1875 

O.II94 

0.2387 

0.1381 

0.2575 

H 

9.4248 

0.1666 

0.1061 

O.2I22 

0.1228 

0.2289 

Me 

10.0531 

0.1562 

0.0995 

0.1989 

0.1151 

0.2146 

9* 

10.9956 

0.1429 

o  .  0909 

O.l8l9 

0.1052 

0.1962 

N 

12.5664 

O.I25O 

0.0796 

O.I59I 

0.0921 

0.1716 

^ 

14.1372 

O.IIII 

0.0707 

O.I4I5 

0.0818 

0.1526 

M 

15  .  7080 

O.IOOO 

0.0637 

0.1273 

0.0737 

o.i373 

Me 

16.7552 

0.0937 

0.0597 

O.II94 

o  .  0690 

0.1287 

N 

18.8496 

0.0833 

0.0531 

0.1061 

0.0614 

0.1144 

M 

21.9911 

o  0714 

0.0455 

0.0910 

0.0526 

0.0981 

H 

25.1327 

0.0625 

o  .  0398 

0.0796 

o  .  0460 

0.0858 

H 

28.2743 

0.0555 

0.0354 

0.0707 

o  .  0409 

0.0763 

Ho 

3I-4I59 

o  .  0500 

0.0318 

0.0637 

0.0368 

0.0687 

He 

50.2655 

0.0312 

0.0199 

0.0398 

0.0230 

0.0429 

From  MACHINERY'S  HANDBOOK,  page  553. 


24  SPUR  GEARING 

spur  gears  with  either  diametral  or  circular  pitch.  Nos.  17  to 
22  give  additional  formulas  for  various  conditions  of  known  and 
unknown  factors.  Formulas  23  and  24  give  the  length  of  a  rack 
when  the  number  of  teeth  in  it  and  either  the  diametral  or  the 
circular  pitch  are  known.  In  the  formulas  in  this  chart,  the 
following  notation  has  been  used: 

P  =  diametral  pitch;  Pf  =  circular  pitch; 

N  =  number  of  teeth;    (if  D  =  pitch  diameter; 

the  number  of  teeth  C  =  center  distance; 

in   both   gear   and  S  =  addendum; 

pinion  are  referred  F  =  clearance; 

to,    Na   =  number  W=  whole  depth  of  tooth; 

of    teeth    in    gear,  T  =  thickness  of  tooth; 

and  Np  =  number  0  =  outside  diameter  of  gear. 

of  teeth  in  pinion) ; 

Internal  Spur  Gears.  —  As  indicated  by  its  name,  the  internal 
gear  is  one  having  teeth  formed  on  an  interior  pitch  surface 
instead  of  on  an  exterior  one.  Briefly,  and  perhaps  somewhat 
unconventionally  defined,  it  is  an  ordinary  spur  gear  turned 
inside  out.  At  the  right  of  Fig.  9  is  shown  a  sketch  of  such  a 
gear,  meshing  with  a  spur  pinion;  at  the  left  is  shown  a  pair  of 
spur  gears  having  the  same  number  of  teeth  and  the  same  pitch 
as  the  pinion  and  internal  gear  on  the  right.  By  tracing  the 
motion  in  each  figure,  it  will  be  seen  that  internal  action  causes 
the  two  members  to  turn  in  the  same  direction,  while  external 
action  produces  opposite  rotation. 

The  Uses  of  Internal  Gearing.  —  There  are  some  advantages 
incident  to  the  use  of  internal  gears  for  particular  applications, 
as  compared  with  external  gears  of  the  same  pitch  and  number 
of  teeth.  For  one  thing,  an  internal  gear  has  its  teeth  and  that 
of  its  pinion  protected  to  a  very  marked  degree  from  inflicting 
or  receiving  injury,  often  making  the  use  of  a  gear  guard  un- 
necessary, if  the  parts  are  properly  designed  for  that  purpose. 
Owing  to  the  fact  that  the  cylindrical  pitch  surfaces  in  internal 
gearing  have  their  curvature  in  the  same  direction,  the  teeth  of 
the  pinion  approach  and  mesh  with  those  of  its  mate  somewhat 


INTERNAL   GEARS  25 

more  gradually  and  easily  than  when  they  are  meshing  with  an 
external  gear.  This  tends  toward  smoothness  and  quietness  in 
running,  as  well  as  giving  a  slightly  longer  contact  for  each 
tooth.  Another  characteristic  which  is  often  an  advantage  will 
be  seen  from  a  study  of  Fig.  9.  In  each  case  shown  we  have 
gears  of  the  same  pitch  and  number  of  teeth.  The  internal 
gears  evidently  have  a  much  smaller  center  distance  than  the 
external  gears.  This  matter  is  of  importance  when  it  is  neces- 
sary to  transmit  considerable  power  between  shafts  placed 
quite  close  together. 


Fig.  9.     Comparison  between  External  and  Internal  Spur  Gears 

In  contrast  with  the  advantages  just  mentioned,  the  chief 
factor  which  has  limited  the  use  of  the  internal  gear  has  been 
the  difficulty  and  expense  of  making  it.  This  difficulty  has  not 
been  insuperable  for  cast  gearing,  but,  until  the  introduction  of 
recent  processes,  the  cutting  of  internal  teeth  has  been  a  tedious 
and  unsatisfactory  process. 

Rules  for  Designing  Internal  Gearing.  —  Neglecting  for  the 
time  being  the  modifications  which  have  to  be  made  in  the 
standard  proportions  to  avoid  interference,  it  may  be  said  that 
the  usual  thing  to  do  in  designing  internal  gearing  is  to  follow 


26  SPUR  GEARING 

exactly  the  dimensions  of  the  standard  system  as  used  for 
external  spur  gearing.  Practically  the  only  modifications  re- 
quired in  the  rules  given  in  the  table  "  Rules  and  Formulas  for 
Dimensions  of  Spur  Gears,"  are  those  made  necessary  by  the 
fact  that  the  center  distance,  in  internal  gearing,  is  equal  to  the 
difference  between  the  two  pitch  radii,  instead  of  to  their  sum. 
Besides  this,  we  have  of  course  to  reckon  with  the  fact  that  the 
teeth  are  "  turned  inside  out,"  so  that  the  bottom  or  root  diam- 
eter is  larger  than  the  pitch  diameter. 

The  only  new  term  is  "  inside  diameter,"  which  takes  the  place 
of  the  outside  diameter  of  external  spur  gearing.  It  is,  of  course, 
the  inside  diameter  of  the  blank  before  the  teeth  are  cut,  and  it 
is  marked  0  in  Fig.  9.  The  following  are  the  rules  which  must 
be  changed: 

No.  5  will  read:  To  find  the  center  distance,  subtract  the  number 
of  teeth  in  the  pinion  from  the  number  of  teeth  in  the  gear  and  divide 
the  remainder  by  2  times  the  diametral  pitch. 

No.  6  will  read:  To  find  the  center  distance,  multiply  the 
difference  of  the  numbers  of  teeth  in  the  gear  and  pinion  by  the 
circular  pitch  and  divide  the  product  by  6.2832. 

No.  15  will  read:  To  find  the  inside  diameter,  subtract  2  from 
the  number  of  teeth  and  divide  the  remainder  by  the  diametral  pitch. 

No.  1 6  will  read:  To  find  the  inside  diameter,  subtract  2  from 
the  number  of  teeth,  multiply  the  remainder  by  the  circular  pitch, 
and  divide  the  product  fry  3.1416. 

No.  19  will  read:  To  find  the  pitch  diameter,  add  twice  the 
addendum  to  the  inside  diameter. 

No.  22  will  read:  To  find  the  inside  diameter,  subtract  twice  the 
addendum  from  the  pitch  diameter. 

Interference  in  Internal  Gears.  —  In  laying  out  the  shape  of 
teeth  for  internal  gearing  we  have  to  look  out  for  two  kinds  of 
interference  which  are  almost  sure  to  be  met  with.  The  points 
of  the  rack  teeth  in  the  i4§-degree  involute  system  are  relieved 
to  avoid  the  interference  with  the  flanks  of  small  pinions,  and 
the  points  of  internal  gear  teeth  have  to  be  relieved  for  the  same 
reason,  but  to  an  even  greater  degree.  A  second  form  of  inter- 
ference occurs  when  the  pinion  has  too  nearly  the  same  number 


GRANT'S  ODONTOGRAPH 


of  teeth  as  the  gear.  In  this  case  there  is  a  tendency  for  the 
points  of  the  pinion  and  the  gear  teeth  to  strike  as  they  roll  into 
and  out  of  engagement. 

For  the  first  form  correction  may  be  made  either  by  correcting 
the  points  of  the  internal  gear  tooth  by  shortening  them  (at  the 
same  time  preferably  lengthening  the  addendum  of  the  pinion), 
or  by  changing  the  pressure  angle.  The  mechanic  who  desires 

Grant's  Odontograph  Table  for  Cycloidal  Teeth  * 


R,  r,  D  and  d  for  One  Diam- 

R, r,  D  and  d  for  One  Inch 

Number  of 

etral  Pitch;  for  any  other 

Circular  Pitch;  for  any  other 

Teeth  in  the 

Pitch  divide  Values  given 

Pitch  multiply  Values  given 

Gear 

by  that  Pitch 

by  that  Pitch 

Faces 

Flanks 

Faces 

Flanks 

Also 

TTv-a/vf 

TT__J 

Jj/XciCt 

used 
for 

R 

D 

r 

d 

R 

D 

r 

4 

IO 

IO 

1.99 

O.O2 

-  8.00 

4.00 

0.62 

O.OI 

-2-55 

1.27 

II 

II 

2.OO 

O.O4 

—  i  i  .  05 

6.50 

0.63 

0.01 

-3-34 

2.07 

12 

12 

2.OI 

O.O6 

OO 

00 

0.64 

0.02 

00 

OO 

I3W 

13-    14 

2.04 

O.O7 

15.10 

9-43 

0.65 

O.O2 

4.80 

3.00 

*$H 

15-  16 

2.IO 

O.O9 

7.86 

3.46 

0.67 

0.03 

2.50 

I.IO 

I7H 

17-  18 

2.14 

O.II 

6.13 

2.  2O 

0.68 

O.O4 

i.  95 

0.70 

20 

19-    21 

2.  2O 

0.13 

5-12 

i-57 

0.70 

O.O4 

•63 

0.50 

23 

22-    24 

2.26 

0-15 

4.50 

1-13 

0.72 

0.05 

•43 

0.36 

27 

25-    29 

2  33 

0.16 

4.10 

0.96 

0.74 

0.05 

•30 

0.29 

33 

30-   36 

2.40 

0.19 

3.80 

0.72 

0.76 

0.06 

.20 

0.23 

42 

37-  48 

2.48 

O.22 

3-52 

0.63 

0.79 

0.07 

.12 

O.2O 

58 

49-  72 

2.60 

0.25 

3-33 

0-54 

0.83 

0.08 

.06 

0.17 

97 

73-144 

2.83 

0.28 

3-14 

0.44 

0.90 

0.09 

.OO 

0.14 

290 

145-300 

2.92 

0.31 

3.00 

0.38 

0-93 

0.10 

o-95 

O.I2 

00 

Rack 

2.96 

0-34 

2.96 

o-34 

0.94 

O.II 

0.94 

O.II 

*  From  "  A  Treatise  on  Gear  Wheels,"  by  George  B.  Grant. 

to  use  internal  cut  gearing  without  making  a  study  of  the  theo- 
retical conditions  has  two  courses  open  to  him.  He  may  pur- 
chase a  formed  cutter  for  the  gear  from  the  regular  makers  of 
formed  cutters,  telling  them  the  number  of  teeth  and  pitch  he 
proposes  to  use  for  the  gear  and  pinion.  In  that  case,  the  maker 
of  the  cutter  will  make  such  corrections  in  its  form  as  may  be 
necessary  to  avoid  interference.  Another  way  is  to  cut  the 
internal  gear  on  the  Fellows  gear  shaper.  With  this  machine 
the  cutter  forms  its  own  correction,  so  that  no  calculation  on  the 
part  of  the  user  is  ordinarily  required. 


28 


SPUR   GEARING 


Grant's  Odontograph.  —  The  tables  of  Grant's  odontographs 
for  cycloidal  and  involute  teeth  provide  a  simple  means  for 
laying  out  gear  teeth  or  templets  for  gear  teeth  accurately  by 
means  of  circular  arcs  which  very  closely  approximate  the  exact 
tooth  curves.  This  method  was  devised  by  Mr.  George  B. 
Grant  and  constitutes  one  of  the  best  methods  known  for  ap- 
proximating the  exact  shapes  of  gear  teeth  by  means  of  circular 
arcs. 

Odontograph  Table  for  the  Cycloidal  System.  —  To  apply  the 
odontograph  table  for  the  cycloidal  system  of  gear  teeth,  first 


LINE     OF     FLANK 


Machinery 


Fig.  10.     Cycloidal  System  Gear  Teeth  laid  out  by  the  Aid  of  Grant's 

Odontograph 

draw  the  pitch,  addendum,  root  and  clearance  circles,  as  indi- 
cated in  the  engraving,  Fig.  10,  and  space  off  the  pitch  of  the 
teeth  on  the  pitch  circle  in  the  usual  way.  Then  draw  the  circle 
marked  "line  of  flank  centers"  at  the  distance  d,  as  given  in  the 
table,  outside  of  the  pitch  line,  and  draw  the  "line  of  face  centers  " 
at  the  distance  D  inside  of  it.  With  the  face  radius  R  in  the 
dividers,  draw  in  all  the  face  curves  from  centers  located  on 
the  "line  of  face  centers."  Then  with  a  flank  radius  r,  draw 
all  the  flank  curves  from  centers  located  on  the  "line  of  flank 
centers." 
The  table  gives  the  distances  D  and  d  and  radii  R  and  r  for 


GRANT'S  ODONTOGRAPH 


29 


pitches  either  exactly  one  diametral  or  one-inch  circular  pitch. 
For  any  other  pitch,  divide  or  multiply  as  directed  in  the  table. 
The  illustration,  Fig.  10,  shows  the  method  applied  to  laying 
out  a  2  diametral  pitch  gear.  The  odontograph  for  the  cycloidal 
system  may  also  be  applied  to  laying  out  teeth  for  internal 
gears. 

Odontograph  Table  for  The  Involute  System.  —  To  draw  the 
tooth  curves,  first  lay  off  the  pitch,  addendum,  root  and  clearance 
circles,  and  space  off  the  teeth  on  the  pitch  line  as  indicated  in 
Fig.  ii.  Draw  the  "base  line"  one-sixtieth  of  the  pitch  diameter 


Machinery 


Fig.  ii.     Involute  System  Gear  Teeth  laid  out  by  the  Aid  of  Grant's 
Odontograph 

inside  the  pitch  line;  then  draw  the  faces  of  the  teeth  from  the 
pitch  line  to  the  addendum  line  by  using  the  face  radius  as  given 
in  the  table  for  involute  teeth,  and  with  centers  located  on  the 
"base  line."  If  the  pitch  is  any  other  than  one  diametral  or  one- 
inch  circular  pitch,  divide  or  multiply  the  values  given  in  the 
table  as  directed.  To  draw  the  flanks  of  the  teeth  from  the  pitch 
line  to  the  base  line,  use  the  flank  radius  given  with  the  centers 
on  the  base  line;  then  draw  straight  radial  flanks  from  the  base 
line  to  the  root  line  and  round  them  off  into  the  clearance  line. 
The  illustration,  Fig.  ii,  shows  the  method  applied  to  laying  out 
a  2  diametral  pitch  gear. 


SPUR   GEARING 


The  odontograph  table  for  involute  teeth  can  be  used  for  in- 
ternal gears  in  the  same  way  as  for  external  gears,  but  care  must 
be  taken  that  the  tooth  of  the  gear  is  cut  off  to  avoid  interference. 
In  fact,  the  point  of  the  tooth  may  be  left  off  altogether,  or 
rounded  off.  The  pinion  tooth  need  not  be  carried  in  to  the 
usual  root  line,  but  may  just  clear  the  truncated  tooth  of  the  gear. 
No  correction  for  interference  is  needed  on  the  points  of  the 
pinion  teeth  or  on  the  flanks  of  the  gear  teeth. 

Grant's  Odontograph  Tables  for  Involute  Teeth  * 


Radii  for  One 

Radii  for  One 

Radii  for  One 

Radii  for  One 

Diametral 

Inch  Circular 

Diametral 

Inch  Circular 

Pitch;  for  any 

Pitch;  lor  any 

Pitch;  for  any 

Pitch;  for  any 

other  Pitch 

other  Pitch 

other  Pitch 

other  Pitch 

No.  of 
Teeth 
in  the 

divide  Values 
given  by  that 
Pitch 

multiply  Values 
given  by  that 
Pitch 

No.  of 
Teeth 
in  the 

divide  Values 
given  by  that 
Pitch 

multiply  Values 
given  by  that 
Pitch 

Gear 

Gear 

Face 

Flank 

Face 

Flank 

Face 

Flank 

Face 

Flank 

Radius 

Radius 

Radius 

Radius 

Radius 

Radius 

Radius 

Radius 

IO 

2.28 

0.69 

0-73 

0.22 

28 

3-92 

2-59 

•25 

0.82 

II 

2.40 

0.83 

0.76 

0.27 

29 

3-99 

2.67 

.27 

0.85 

12 

2.51 

0.96 

0.8o 

0.31 

30 

4.06 

2.76 

.29 

0.88 

13 

2.62 

.09 

0.83 

0-34 

31 

4-13 

2.85 

•31 

0.91 

14 

2.72 

.22 

0.87 

0-39 

32 

4.20 

2-93 

•34 

0-93 

IS 

2.82 

•34 

0.90 

o-43 

33 

4.27 

3-01 

•36 

0.96 

16 

2.  Q2 

.46 

0-93 

0.47 

34 

4-33 

3-09 

1.38 

0.99 

17 

3.02 

.58 

0.96 

0.50 

35 

4-39 

3-l6 

1-39 

i  .01 

18 

3-12 

.69 

0.99 

0-54 

36 

4-45 

3-23 

1.41 

1.03 

J9 

3.22 

•  79 

1.03 

0-57 

37-  40 

4.20 

i-34 

20 

3-32 

.89 

i.  06 

0.60 

4i-  45 

4-63 

1.48 

21 

3-41 

.98 

.09 

0.63 

46-  51 

5-o6 

1.61 

22 

3-49 

2.06 

.11 

0.66 

52-  60 

5-74 

1-83 

23 

3-57 

2-15 

•13 

0.69 

6  1-  70 

6.52 

2.07 

24 

3.64 

2.24 

.16 

0.71 

71-  90 

7.72 

2.46 

25 

3-71 

2-33 

.18 

0.74 

91-120 

9.78 

3.H 

26 

3-78 

2.42 

.20 

0.77 

121-180 

13-38 

4.26 

2? 

3-85 

2.50 

•23 

0.80 

181-360 

21.62 

6.88 

*  From  "  A  Treatise  on  Gear  Wheels,"  by  George  B.  Grant. 

Special  Rule  for  Involute  Rack.  —  Draw  the  sides  of  the  rack 
teeth  as  straight  lines  inclined  14 J  degrees  to  the  center  line  cOc, 
Fig.  ii.  Draw  the  outer  half  ab  of  the  face  by  means  of  a  circu- 
lar arc  having  a  radius  of  2.10  inches  divided  by  the  diametral 
pitch,  or  0.67  inch  multiplied  by  the  circular  pitch,  the  center  for 
this  arc  being  on  the  pitch  line  of  the  rack. 


GEAR  CUTTERS 


Cutters  for  Involute  and  Cycloidal  Teeth.  —  According  to  the 
system  for  cutting  gear  teeth  adopted  by  the  Brown  &  Sharpe 
Mfg.  Co.,  Providence,  R.  I.,  any  gear  of  one  pitch  will  mesh  with 
any  other  gear  or  with  a  rack  of  the  same  pitch.  Eight  cutters 
are  required  for  each  pitch.  These  eight  cutters  are  adapted  to 
cut  from  a  pinion  of  twelve  teeth  to  a  rack,  and  are  numbered, 
respectively,  i,  2,  3,  etc.  The  number  of  teeth  and  the  pitch  for 
which  a  cutter  is  adapted  is  always  marked  on  each.  A  list  of 
these  cutters  is  given  in  Table  I. 

Cutters  for  the  cycloidal  form  of  teeth  are  also  made  so  that 
any  gear  of  one  pitch  will  mesh  into  any  other  gear  or  into  a  rack 
of  the  same  pitch,  but  twenty-four  cutters  are  required  for  each 
pitch.  In  order  that  gears  with  this  form  of  teeth  shall  run  well 
together,  they  must  be  cut  accurately  to  the  required  depth; 

Table  I.    Cutters  for  Involute  Gear  Teeth 


No.  of 
Cutter 

Number  of 
Teeth 

No.  of 
Cutter 

Number  of 
Teeth 

No.  of 
Cutter 

Number  of 
Teeth 

I 
2 
2 

135  to  rack 
55  to  134 
l<  to  t?4 

4 

I 

26  to  34 
21  to  25 
17  to  20 

8 

14  to  16 

12  tO  13 

otherwise  the  pitch  circles  will  not  be  tangent  to  each  other.  To 
secure  a  proper  depth  of  tooth,  the  cutters  are  made  with  a 
shoulder  which  determines  the  exact  depth  that  the  tooth  should 
be  cut.  Thus,  if  care  is  taken  when  turning  the  blanks,  to  obtain 
the  correct  outside  diameter  of  the  gear,  no  measurements  need 
be  taken  when  cutting  the  teeth.  The  twenty-four  cutters  are 
adapted  to  cut  from  a  pinion  of  twelve  teeth  to  a  rack,  and  are 
designated  by  letters  A}  B,  C,  etc.  The  number  of  teeth  and  the 
pitch  for  which  the  cutter  is  adapted  is  always  marked  on  each, 
the  same  as  in  the  case  of  cutters  for  involute  teeth.  A  list  of 
these  cutters  is  given  in  Table  II. 

Importance  of  Grinding  Gear-cutter  Teeth  Radially.  —  Fig.  12 
shows,  diagrammatically,  how  the  teeth  of  a  milling  cutter  for 
gear  teeth  should  be  ground  to  secure  the  best  results;  it  also 
illustrates  improper  grinding.  The  teeth  A  and  B  are  ground 


SPUR  GEARING 


correctly.  The  lines  AC  and  BC,  lying  in  the  plane  of  the  cut- 
ting face,  are  radial;  that  is,  the  faces  of  the  teeth  would  pass 
directly  through  the  center  of  the  cutter,  if  projected  to  the 
center.  Tooth  D,  however,  shows  an  entirely  different  condi- 
tion, and  one  which  unfortunately  is  not  uncommon  in  gear- 
cutting  practice.  The  top  of  the  tooth  was  ground  back  faster 
than  the  base,  thus  throwing  the  face  of  the  cutter  into  the  plane 
indicated  by  the  line  DE'}  consequently  the  shape  of  the  tooth 
space  cut  is  distorted,  and  a  gear  with  badly-shaped  teeth  must 
necessarily  be  produced  by  it. 

The  expression,  "may  be  ground  without  changing  the  form," 
is  often  taken  too  literally  and  without  the  required  qualifica- 

Table  H.     Cutters  for  Cycloidal  Gear  Teeth 


Letter  of 
Cutter 

Number  of 
Teeth 

Letter  of 
Cutter 

Number  of 
Teeth 

Letter  of 
Cutter 

Number  of 
Teeth 

A 

12 

/ 

20 

0 

43  to    49 

B 

13 

/ 

21  to  22 

R 

50  to    59 

C 

14 

K 

23  to  24 

S 

60  to    74 

D 

15 

L 

25  to  26 

T 

75  to    99 

E 

16 

M 

27  to  29 

U 

100  to  149 

F 

17 

N 

30  to  33 

V 

150  to  249 

G 

18 

O 

34  to  37 

W 

250  or  more 

H 

iQ 

P 

38  to  42 

X 

Rack' 

tion  that  it  is  necessary  to  grind  in  a  plane  radial  with  the  center 
of  the  cutter  in  order  that  the  form  shall  not  be  changed.  It  is 
evident  to  anyone  who  will  give  the  matter  a  little  thought  that 
if  a  gear  is  cut  with  a  gear-cutter  having  teeth  ground  like  D  the 
resulting  tooth  space  will  be  too  wide  at  the  top,  if  the  cutter  is 
carried  to  the  correct  depth.  Moreover,  such  a  gear-cutter 
works  badly,  as  the  cutting  faces  of  the  teeth  have  a  negative 
rake.  The  importance  of  correct  grinding  of  all  formed  cutters 
can,  therefore,  not  be  too  strongly  emphasized.  Unfortunately, 
formed  cutters  that  can  be  ground  without  changing  the  form 
do  not  always  have  sufficient  clearance  to  work  well  with  all 
classes  of  work,  and  if  such  cutters  are  carelessly  used  there  will 
be  heating  and  rapid  wearing  away  of  the  tops  of  the  teeth.  If 
hard  pressed  and  ignorant,  the  tendency  of  the  grinding  operator, 


GEAR   CUTTERS 


33 


in  order  to  hurry  the  sharpening  of  such  cutters,  is  to  incline  the 
wheel  away  from  the  radial  plane. 

Chordal  Thicknesses  and  Addenda  for  Gear  Teeth  and  Gear 
Cutters.  —  In  measuring  the  thickness  of  gear  teeth  and  gear 
cutters,  it  is  necessary  to  make  allowance  for  the  curve  of  the 
pitch  circle  of  the  gear.  In  the  following  will  be  given  formulas 
for  finding  the  chordal  thicknesses  and  what  is  called  the  "cor- 
rected" addenda,  that  is,  the  perpendicular  distance  from  the 


TOOTH  INCORRECTLY 

GROUND, 
FACE  OF  TOOTH  NOT  RADIAL 


TH  CORRECTLY 

GROUND, 
f~\CE  OF  TOOTH  RADIAL 


Machinery 


Fig.  12.     Correct  and  Incorrect  Methods  of  Grinding  Gear-cutter  Teeth 

chord  at  the  pitch  circle  to  the  outside  diameter  of  the  gear  as 
indicated  at  H,  in  Fig.  13.     Let, 

a  =  half  the  angle  subtended  from  the  center  of  the  gear  by 
one  gear  tooth  (see  Fig.  13); 

N  =  number  of  teeth  in  gear; 

T  =  chordal  thickness  of  tooth  at  pitch  line; 

D  =  perpendicular  distance  from  chord  T  to  center  of  gear; 

H  =  perpendicular  distance  from  chord  to  outside  circum- 
ference of  gear; 

C  =  radius  of  gear  at  pitch 

E  =  outside  radius  of  gear. 


34 


SPUR  GEARING 


The  formulas  are  as  follows 


a  = 


N 
D  =  VC^ 


T  =  2C  Xsina; 


T)2;    H  =  E-D. 

In  the  case  of  the  gear  cutter  (see  Fig.  14),  the  chordal  thick- 
ness is  the  same  as  that  for  the  gear,  but  the  corrected  addendum 
of  the  gear  cutter  is  different  from  the  corrected  addendum  of 
the  gear.  The  two  dimensions,  however,  added  together  must 
equal  the  total  depth  of  the  gear  tooth.  To  obtain  the  corrected 


I 

Machine? 


Figs.  13  and  14.     Notation  used  in  Formulas  for  Chordal  Thicknesses 
and  Addenda  of  Gear  Teeth  and  Cutters 

addendum  A  of  the  gear  cutter,  we  can,  therefore,  either  sub- 
tract the  dimension  H,  as  found  by  the  previous  formulas,  from 
the  dimension  for  the  total  depth  of  the  tooth,  or  we  can  take 
the  dedendum  for  the  particular  pitch  required  from  any  stand- 
ard table  of  gear  tooth  parts  and  subtract  the  dimension  K, 
Fig.  14,  which  is  found  by  the  formula: 
K  =  C  (i  —  cos  a). 

Testing  the  Tooth  Thickness  when  Milling  Gear  Teeth.  — 

The  special  vernier  gear- tooth  caliper  illustrated  in  Fig.  15  is 
sometimes  used  for  testing  the  thickness  of  the  first  tooth  milled. 


MODULE  SYSTEM  OF   GEAR  TEETH 


35 


This  test  is  especially  desirable  if  there  is  any  doubt  about  the 
accuracy  of  the  blank  diameter.  To  test  the  tooth  thickness, 
a  trial  cut  is  taken  for  a  very  short  distance  at  one  side  of  the 
blank;  then  the  work  is  indexed  for  the  next  space,  after  which 
another  trial  cut  is  taken  part  way  across  the  gear.  The  vertical 
scale  of  the  caliper  is  set  so  that  when  it  rests  on  top  of  the  tooth, 
as  shown,  the  lower  ends  of  the  caliper  jaws  will  be  at  the  height 
of  the  pitch  circle.  The  horizontal  scale  then  shows  the  chordal 


Machinery 


Fig.  15. 


Vernier  Caliper  for  Measuring  Thickness  of  Gear  Teeth  at 
the  Pitch  Circle 


thickness  of  the  tooth  at  this  point.  When  a  gear  tooth  is 
measured  in  this  way,  it  is  the  chordal  thickness  T  (see  Fig.  13) 
that  is  obtained,  instead  of  the  thickness  along  the  pitch  circle, 
as  explained  in  the  previous  paragraph.  Hence,  when  measur- 
ing teeth  of  coarse  pitch,  especially  if  the  diameter  of  the  gear 
is  quite  small,  dimension  T  should  be  obtained.  It  is  also 
necessary  to  obtain  the  corrected  addendum  fl",  in  order  to 
measure  the  chordal  thickness  T  at  the  proper  point  on  the  sides 
of  the  tooth. 


SPUR  GEARING 


Limits  for  Gearing.  —  The  limits  for  center  distance,  pitch 
diameter  and  outside  diameter  of  blanks,  given  in  the  table 
below,  are  applicable  to  spur  gearing  used  under  ordinary  con- 
ditions. The  +  sign  indicates  dimensions  over,  and  the  — 
sign,  dimensions  under,  the  actual  theoretical  dimension. 

Manufacturing  Limits  for  Gearing 


Diametral 
Pitch 

Center 
Distance 

Pitch  Diameter 

Blanks,  Outside 
Diameter 

16 

±O.OO2 

—0.003    to  —0.005 

o.ooo  to  —0.005 

14 

±0.003 

—0.004    to  —0.006 

o.ooo  to  —0.005 

12 

±0.0035 

—0.0045  to  —0.007 

o.ooo  to  —0.006 

IO 

±0.004 

—0.005    to  —0.008 

o.ooo  to  —0.006 

8 

±0.005 

—0.006    to  —0.009 

o.ooo  to  —0.007 

6 

±0.006 

—0.007    to  —  o.oio 

o.ooo  to  —0.008 

5 

±0.007 

—0.008    to  —  o.on 

o.ooo  to  —o.oio 

4 

±0.008 

—  O.OOQ     tO  —  O.OI2 

o.ooo  to  —0.015 

Metric  or  Module  System  of  Gear  Teeth.  —  In  the  metric 
system,  the  diametral  pitch  is  not  used,  but  instead,  the  dimen- 
sions of  gear  teeth  are  expressed  by  reference  to  the  module  of 
the  gear.  The  module  is  equal  to  the  pitch  diameter  in  milli- 
meters divided  by  the  number  of  teeth  in  the  gear.  For  example, 

Table  of  Tooth  Parts  for  Metric  or  Module  System  Gear  Teeth 


Module 

Circular 
Pitch, 
Inches 

Thickness  of 
Tooth  at  Pitch 
Line, 
Inches 

Addendum, 
Inches 

Whole  Depth 
of  Tooth, 
Inches 

tH 

0.1546 

0.0773 

0.0492 

0.1061 

iH 

0.1854 

0.0927 

0.0591 

0.1273 

r« 

0.2164 

o.  1082 

0.0686 

0.1486 

2 

0.2472 

0.1236 

0.0787 

0.1698 

2H 

0.2784 

0.1392 

0.0886 

0.1911 

2H 

0.3092 

0.1546 

o  .  0984 

0.2123 

2% 

o  .  3402 

O.I70I 

0.1083 

0.2336 

3 

0.3710 

0-1855 

o.  1181 

0.2548 

3H 

0.4330 

0.2165 

0.1378 

0.2973 

4 

0.4948 

0-2474 

0.1575 

0.3397 

4H 

0.5568 

0.2784 

0.1772 

0.3822 

5 

0.6186 

0.3093 

0.1969 

0.4247 

5H 

0.6802 

0-3401 

0.2165 

0.4670 

6 

0.7420 

0.3710 

0.2362 

0.5095 

MODULE   SYSTEM   OF   GEAR  TEETH 


37 


if  the  pitch  diameter  of  a  gear  is  50  millimeters  and  the  number 
of  teeth,  25,  then  the  module  equals  50  -f-  25  =  2.     The  accom- 

Module  and  Corresponding  Diametral  Pitch 


Corre- 

Corre- 

Corre- 

Corre- 

sponding 

sponding 

sponding 

sponding 

Module 

English 

Module 

English 

Module 

English 

Module 

English 

Diametral 

Diametral 

Diametral 

Diametral 

Pitch 

Pitch 

Pitch 

Pitch 

0-5 

50.800 

2.25 

11.288 

5-0 

5.080 

II 

2  309 

0-75 

33.867 

2-5 

10.  160 

5-  5 

4.618 

12 

2.II7 

I  .O 

25.400 

2-75 

9.236 

6.0 

4.233 

14 

1.814 

1-25 

20.320 

3-0 

8.466 

7.0 

3-628 

16 

1.587 

i-5 

16.933 

3-5 

7-257 

8.0 

3-175 

18 

1.411 

1-75 

14.514 

4.0 

6.350 

9.0 

2.822 

20 

1  .270 

2.0 

12.700 

4-5 

5.644 

IO.O 

2.540 

24 

1.058 

panying  table  gives  a  comparison  between  the  module  and  the 
corresponding  diametral  pitch  of  gears. 

The  module  is  also  equal  to  the  circular  pitch  in  millimeters 
divided  by  3.1416.     Either  rule  gives  the  same  result. 


CHAPTER  II 
MATERIALS  USED   FOR  GEARS 

THE  principal  materials  used  for  gears  are  cast  iron,  steel  (both 
from  bar  stock  and  in  the  form  of  forgings  or  castings),  brass  and 
bronze  (from  stock  or  castings),  rawhide,  and  fiber. 

Cast  Iron.  —  Cast  iron  is,  perhaps,  the  most  used  material  for 
gears.  It  is  one  of  the  cheapest,  and  is  the  easiest  to  mold  or  cut 
to  shape.  It  wears  fairly  well  also.  Its  greatest  disadvantages 
are  its  lack  of  resisting  power  to  shock  or  impact,  and  the  un- 
certainty of  the  quality  of  the  casting  into  which  it  is  formed.  A 
casting  is  liable  to  various  more  or  less  serious  defects,  some  of 
which  may  be  visible  from  the  exterior,  while  others  are  con- 
cealed; "blow-holes,"  "cold  shuts,"  "scabs,"  etc.,  are  of  com- 
mon occurrence  where  the  foundry  work  is  not  skillfully  done. 
Castings  from  iron  are  so  cheap,  however,  that  those  containing 
such  defects  may  be  discarded  without  hesitation  as  soon  as  they 
are  discovered.  The  prime  advantage  of  the  material,  aside 
from  its  cheapness,  is  the  facility  with  which  it  may  be  molded 
into  any  desired  form,  so  that  we  may  have  large  gears  with  arms, 
webs,  projecting  bosses,  counterbalances,  etc.,  to  suit  the  mech- 
anism being  designed.  These  advantages  will  doubtless  con- 
tinue to  keep  cast  iron  the  most  used  of  all  materials  for  ordinary 
work. 

Cast  Steel.  —  When  greater  strength  than  that  obtainable 
in  a  cast-iron  gear  is  required,  and  the  gear  is  too  large  to  be 
made  from  blanks  cut  from  round  stock  or  from  forgings,  steel 
castings  are  frequently  used  for  gears.  This  is  considered  the 
best  material  for  large,  heavy-duty  gears.  The  art  of  making 
castings  from  steel  so  that  they  will  be  sound  throughout  has, 
however,  only  become  understood  in  the  past  few  years,  and 
blow-holes  and  rough  castings  are  common  enough,  so  that  the 
use  of  this  material  still  labors  under  some  disadvantages,  except 
in  cases  where  the  foundries  are  well  equipped  for  doing  this 

38 


MATERIALS  FOR  GEARS  39 

class  of  work,  when  it  is  possible  to  turn  out  a  first-class  product. 
In  the  design  of  gears  from  cast  steel,  it  seems  that  true  economy 
is  frequently  lost  sight  of,  for  although  cast  steel  is  by  far  the 
stronger  material,  the  arms,  hubs  and  rims  of  cast-steel  gears  for 
similar  duty  are  often  made  of  the  same  proportions  as  those  for 
cast-iron  gears. 

Comparison  between  Cast  Iron  and  Cast  Steel  for  Gears.  — 
By  far  the  greatest  number  of  all  gears  are  made  from  cast  iron  or 
cast  steel.  Concerning  the  relative  advantages  of  these  two 
materials  there  is  much  difference  of  opinion.  Of  late  years, 
users  of  gears  seem  to  have  accepted,  without  qualifications,  the 
idea  that  for  large  gears  cast  steel  is  the  best  material  under  any 
and  all  conditions,  but  nothing  could  be  more  erroneous  than 
this,  because  under  all  ordinary  conditions  cast-steel  gears  have 
but  a  single  sharply  defined  superiority  over  cast-iron  gears,  and 
that  is  greater  strength;  and  while  this  is  the  main  advantage, 
this  fact  has  not  been  clearly  apprehended  by  designers,  because 
as  already  mentioned,  no  appreciable  reduction  has  been  made 
in  the  size  of  the  hubs,  arms  or  rims  of  gears,  on  account  of  the 
fact  that  they  are  made  of  steel  —  the  cast-iron  sizes  still  pre- 
vailing in  most  cases. 

On  the  other  hand,  cast  iron  has  a  number  of  distinct  ad- 
vantages over  cast  steel,  both  from  the  designer's  and  the  user's 
point  of  view.  Cast  iron  is  a  much  easier  metal  to  work  with 
than  cast  steel,  having  less  shrinkage  and  less  warpage  than  the 
latter ;  nor  will  it  so  readily  transmit  vibration.  This  latter  pecu- 
liarity can  readily  be  demonstrated  by  striking  with  a  hammer 
on  a  wheel  of  cast  steel  and  on  one  of  cast  iron.  The  tone  of 
the  resulting  noise  is  much  higher  in  pitch  in  the  case  of  the  cast 
steel,  due  to  the  shorter  and  more  rapid  vibrations.  The  noise 
from  the  cast  iron  is  lower  in  tone  and  of  shorter  duration,  due  to 
the  opposite  conditions.  Thus  cast-iron  gears  immediately  take 
the  preference  from  the  standpoint  of  closest  approach  to  silence 
in  operation.  Only  in  exceptional  cases  is  it  necessary  to  use 
cast-steel  gears  for  their  superior  strength.  In  the  vast  majority 
of  cases  a  good  cast-iron  gear  is  amply  strong  enough  for  the 
service  required  of  it. 


40  SPUR  GEARING 

Comparison  between  Cast-iron  and  Cast-steel  Gears  with 
Relation  to  the  Methods  of  Production.  —  There  are  three  con- 
ventional methods  of  making  gears.  When  the  teeth  are  to  be 
cut,  the  rim  of  the  pattern  is  made  a  solid  blank.  The  resulting 
casting  is  bored,  turned  on  its  periphery  and  faced  on  both  sides 
of  the  rim.  The  finished  blank  is  then  put  on  the  gear-cutting 
machine  and  the  teeth  cut  as  desired.  The  two  remaining 
methods  relate  to  gears  designed  to  have  cast  teeth.  In  the  one 
case  a  full  pattern  is  made;  that  is,  all  the  teeth  are  cut  on  the 
pattern.  This  is  the  old-fashioned,  expensive  way  which  entails 
a  vast  deal  of  labor  both  in  the  laying  out  and  the  formation  of 
the  teeth.  Although  tooth-forming  machines  have  done  much 
to  relieve  this  situation  in  the  jobbing  shops,  the  spacing  of  the 
formed  teeth  on  the  periphery  of  the  wheel  is  extremely  difficult 
to  do  accurately  —  simple  as  it  looks  to  those  who  have  never 
had  to  do  it. 

The  other  method  referred  to  is  that  of  machine-molding  the 
gear.  This  is  by  far  the  best  method,  being  the  simplest,  most 
accurate  and  the  cheapest.  All  that  is  required  in  the  way  of  a 
pattern  is  a  sweep,  a  part  of  the  hub,  an  arm,  and  a  tooth  block. 
The  latter  is  fixed  on  the  arm  of  the  machine  and  two  or  three 
teeth,  as  may  be  desired,  are  rammed  at  a  time,  the  spacing  being 
done  with  absolute  precision  by  the  machine.  The  clearances 
in  a  well-made  machine-molded  gear  may  be  nearly  as  close  as  in 
a  cut  gear  of  the  same  pitch.  The  comparative  ease  with  which 
machine-molded  iron  gears  can  be  made  makes  them  by  far  the 
most  economical  to  use  for  larger  sizes,  where  extreme  precision 
is  not  required,  and  if  this  fact  were  better  known  to  engineers 
and  designers  of  machinery,  cast-iron  gears  produced  by  this 
method  would  invariably  be  specified. 

Everyone  familiar  with  foundry  practice  knows  that  the 
outside  skin  of  a  casting  is  its  hardest  part.  This  single  pecu- 
liarity lends  to  machine-molded  cast-iron  gears  the  soundest 
argument  in  their  favor.  It  takes  long  service  to  wear  through 
this  hard  shell  on  the  faces  of  the  teeth,  whereas  in  the  case 
of  cut  gears,  all  of  this  valuable  wearing  surface  is  entirely  cut 
away  before  the  wheels  go  into  service.  Cast-steel  gears  do 


MATERIALS  FOR  GEARS  41 

not  have  these  advantages.  The  high  temperature  at  which 
steel  is  poured  necessitates  the  use  of  silica  sand  in  the  molds. 
This  has  to  be  rammed  so  hard  that  machines  cannot  conveniently 
be  used,  making  a  full  pattern  necessary.  Moreover,  steel 
castings  are  hard  to  make;  it  takes  a  "real  molder"  to  make  a 
good  one.  Since  nearly  all  the  labor  in  the  steel  foundries  is 
done  by  unskilled  men,  who  do  not  appreciate  the  niceties  of 
gear  molding,  it  is  extremely  difficult  to  get  a  good  cast-steel 
gear,  accurate  to  pitch,  round,  and  true  to  theoretical  tooth 
contour.  The  silica  sand  burns  to  the  faces  of  the  teeth,  which 
"scabs"  and  "pits"  them,  so  that  almost  invariably  every  tooth 
face  has  to  be  chipped  to  secure  even  an  indifferent  bearing; 
hence  the  noise  and  clatter  that  is  so  noticeable  in  cast-steel 
gears  with  cast  teeth.  If  the  teeth  do  not  bear  across  their 
entire  face,  the  very  object  that  the  designer  had  in  mind  is 
altogether  lost. 

The  matter  of  shrinkage  in  cast  steel  is  difficult  to  control, 
so  much  so,  in  fact,  that  few  manufacturers  will  guarantee 
large  wheels  to  come  near  the  pattern  allowances.  No  two 
wheels  will  shrink  just  alike,  so  that  the  question  of  whether  the 
wheels  will  preserve  the  circularity  of  the  pattern  is  in  grave 
doubt.  Therefore,  aside  from  exceptional  cases  where  great 
strength  is  required,  designers  should  be  extremely  chary  of 
specifying  cast-steel  gears  unless  the  teeth  are  to  be  cut.  There 
is  but  one  principal  object  in  cutting  the  teeth  of  gear  wheels; 
there  are  a  number  of  minor  reasons.  The  first  is  to  insure 
perfect  tooth  contact;  the  others  are  to  lessen  the  noise  of 
operation,  to  improve  the  appearance  of  the  wheels,  to  discover 
flaws  in  the  material,  etc.;  but  the  expense  of  cutting  is  an 
appreciable  consideration. 

The  argument  is  often  advanced  that  steel  wears  longest. 
The  hard  skin  on  the  teeth  of  machine-molded  gears  of  cast  iron 
so  lengthens  their  time  of  service  that  .even  that  argument  is  in 
grave  doubt.  We  are  brought  face  to  face  with  the  fact  that, 
after  all,  for  a  large  proportion  of  conditions  met  with,  nothing 
can  quite  take  the  place  of  a  well-made  machine-molded  cast- 
iron  gear. 


42  SPUR  GEARING 

Materials  for  Medium-sized  and  Small  Gears.  —  In  a  pair  of 
gears  the  pinion  is  often  made  of  steel,  from  bar  stock  or  a  drop 
forging,  even  when  the  larger  gear  is  made  from  cast  iron  or 
some  other  material.  Steel  has  the  advantage  over  cast  iron  of 
being  more  resilient  —  that  is,  it  offers  a  greater  resistance  to 
shock  or  impact.  Since  the  pinion  of  a  pair  of  gears  naturally 
has  teeth  of  a  weaker  form  than  those  of  its  mate,  it  should  be 
made  of  stronger  material.  Furthermore,  there  is  usually  less 
friction  between  two  different  metals  in  contact  than  between 
two  parts  made  of  the  same  material.  Still  further,  the  smaller 
of  a  pair  of  gears  will  wear  out  much  faster  than  its  mate,  as  each 
of  its  teeth  is  in  action  a  greater  number  of  times  in  a  given 
period;  so  on  this  account  as  well,  it  should  be  made  of  the  more 
resisting  material.  As  the  softer  grades  of  steel,  however,  are 
not  very  durable,  steel  pinions  are  sometimes  casehardened,  or 
they  may  be  made  of  high-carbon t  steel  that  can  be  hardened 
without  requiring  the  action  of  carbonizing  materials.  By  such 
processes  the  pinions  gain  in  durability,  but  suffer  somewhat  in 
accuracy  of  outline,  since  the  natural  result  of  heat-treatment  of 
any  kind  is  to  warp  and  distort  the  part  treated.  It  is  possible 
to  grind  hardened  gears  to  the  correct  outline,  leaving  a  little 
stock  on  the  sides  of  the  teeth,  after  cutting,  for  this  purpose. 
There  would,  however,  seem  to  be  some  doubt  of  the  commercial 
success  of  the  process,  owing  to  the  difficulty  of  keeping  the 
grinding  wheel  to  its  proper  shape,  and  keeping  the  mechanism 
of  the  machine  itself  in  proper  condition  in  the  presence  of  the 
emery  dust  with  which  it  is  surrounded. 

Steel  blanks  for  medium-sized  gears  are  sometimes  drop- 
forged  to  bring  the  wheel  to  the  desired  shape.  Fairly  long  hubs, 
a  thickened  rim,  and  a  thin  web  can  be  formed  in  this  way  with- 
out requiring  the  form  to  be  turned  out  of  solid  metal,  with  the 
attending  waste  of  time  and  material.  The  steel  drop  forging, 
thus,  has  some  of  the  advantages  of  the  casting.  It  is  more  costly 
than  the  casting,  but  may  be  made  of  better  material. 

When  gears  are  made  of  steel  from  the  bar  or  from  forgings,  a 
wide  range  of  physical  qualities  is  offered.  The  steel  may  have 
almost  any  desired  strength,  hardness  and  resilience,  or  capacity 


MATERIALS   FOR  GEARS  43 

to  resist  shock.  The  matter  of  hardness  is  important  in  the  case 
of  high-speed  gearing.  Two  soft  materials  working  on  each 
other  at  a  high  velocity  tend  to  abrade  each  other,  so  one  at  least 
of  a  pair  of  gears  so  used  should  be  of  a  fairly  hard  grade  of  steel. 

Brass  or  Bronze  Gears.  —  It  is  common  and  good  practice  to 
use  a  composition  metal  like  brass  or  bronze  for  the  smaller  of  a 
pair  of  lightly  loaded  gears  which  have  to  run  at  high  speed. 
When  such  gears  are  run  with  a  large  gear  of  cast  iron,  the  differ- 
ence in  texture  between  the  two  materials  used  lessens  the 
friction,  and  there  is  a  gain  on  the  score  of  noiselessness  as  well. 
Brass  may  be  used  where  the  duty  is  very  light;  higher  grades  of 
material,  like  phosphor-bronze,  are  used  for  heavier  service  at 
high  speed.  When  the  service  becomes  quite  severe,  the  materials 
in  the  gears  should  be  reversed,  so  that  the  larger  one  is  of  phos- 
phor-bronze, and  the  smaller  one  of  steel.  The  pinion  has  thus 
its  maximum  of  strength  and  durability,  at  the  same  time  that 
the  advantages  resulting  from  the  use  of  dissimilar  materials  are 
retained. 

Rawhide  Gears.  —  Where  noiselessness  is  a  prime  considera- 
tion, rawhide  is  extensively  used.  This  non-metallic  substance 
possesses  the  required  structure  to  deaden  the  sound  vibrations, 
together  with  a  considerable  degree  of  toughness,  when  properly 
cured.  Manufacturers  of  gear  blanks  from  this  material  cure  the 
hide  by  processes  which  they  claim  give  far  better  results  for  this 
service  than  can  be  obtained  by  ordinary  means.  The  material 
is  not  injured  by  oil,  though  it  does  not  require  lubrication  in 
service;  but  there  has  been  some  complaint  of  its  swelling  and 
losing  its  shape  when  exposed  to  moisture.  Trouble  from  this 
source  may,  however,  have  been  due  to  the  use  of  an  inferior 
grade  of  material,  because  thousands  of  rawhide  pinions  are  in 
satisfactory  daily  use,  under  all  sorts  of  conditions,  at  the  present 
time.  It  is  a  somewhat  more  costly  material  than  the  others 
commonly  used,  but  its  compensating  freedom  from  noise  is  often 
worth  more  than  the  added  expense.  But  one  of  a  pair  of  gears 
—  generally  the  pinion  —  is  made  from  this  substance,  the  gear 
being  of  steel  or  iron.  Gears  as  large  as  40  inches  in  diameter 
have  been  made  from  this  material. 


44  SPUR  GEARING 

Fiber  Gears.  —  Fiber  is  another  material  used  under  about 
the  same  conditions  as  rawhide.  It  is  not  as  strong,  and  it  suffers 
under  the  disadvantage  of  being  difficult  to  machine,  owing  to  its 
peculiar  gritty  structure.  It  is  also  liable  to  swell  in  the  presence 
of  moisture.  It  has  an  advantage  over  rawhide  in  that  it  is  com- 
paratively inexpensive,  and  may  be  purchased  in  a  variety  of  sizes 
of  bars,  rods,  tubes  etc.,  so  that  it  is  convenient  to  use  at  short  no- 
tice. For  light  duty  at  high  speed  it  serves  its  purpose  very  well. 

Cloth  Gears.  —  A  new  material  for  gears  has  been  introduced 
within  recent  years  by  the  General  Electric  Co.;  here  cloth 
pinions  are  employed  in  gear  transmissions  where,  because  of  the 
noise  or  for  other  reasons,  the  meshing  of  metallic  pinions  with 
metallic  gears  would  be  impracticable  or  undesirable.  The  ad- 
vantages claimed  for  these  pinions  are  the  great  tooth  strength, 
the  noiseless  operation,  freedom  from  damage  by  exposure  to 
dampness,  dryness  or  temperature  changes,  the  elasticity  of  the 
teeth,  which  will  absorb  shocks  liable  to  break  cast  iron  or  brass 
pinions,  and  the  long  life  of  the  gears. 

The  blanks  from  which  these  cloth  pinions  are  made  consist  of 
a  filler  of  cotton  or  similar  material  which  is  confined  at  a  pressure 
of  several  tons  to  the  square  inch  between  steel  side  plates  which 
are  held  together  by  threaded  rivets,  or,  in  the  case  of  very  small 
pinions,  by  threaded  sleeves.  After  the  teeth  are  cut,  the  cloth 
filler  is  impregnated  with  oil,  and  becomes  entirely  impervious  to 
moisture  and  unaffected  by  atmospheric  changes.  The  strength 
of  these  cloth  pinions  is  equal  to  that  of  any  other  non-metallic 
gearing,  and  as  the  slight  elasticity  gives  the  meshing  teeth  a 
good  bearing  across  the  full  width  of  the  face,  together  with  good 
shock-absorbing  qualities,  these  gears  can  safely  be  used  for 
practically  any  service  within  the  limits  of  strength  of  cast-iron 
gears.  The  thickness  of  the  side  plates  of  these  pinions  varies 
somewhat  with  the  diameter  and  pitch,  but,  in  general,  it  is 
approximately  equal  to  the  thickness  of  the  teeth  at  the  pitch 
line.  The  width  of  the  cloth  face  is  equal  to  the  width  of  the  gear 
with  which  the  pinion  is  in  mesh,  plus  the  total  end  play  of  both 
shafts,  so  that  the  side  plates  will  not  come  in  contact  with  the 
meshing  gear. 


MATERIALS   FOR  GEARS  45 

Application  of  Cloth  Pinions.  —  These  cloth  pinions  are  in- 
tended for  use  on  machine  tools  such  as  lathes,  planers,  drill 
presses,  shears,  punches,  etc.,  and  are  especially  recommended 
for  motor-driven  tools.  They  are  also  used  on  traveling  cranes 
and  for  driving  looms  and  spinning  frames  and  for  heavy  paper 
and  pulp  mill  machinery,  gas  engine  ignition  drives,  etc.  The 
invention  of  the  cloth  pinion  was  the  result  of  a  search  for  a 
pinion  to  successfully  operate  a  combined  punch  and  shear  used 
in  one  of  the  General  Electric  Co.'s  forge  shops.  This  machine 
was  originally  fitted  with  a  train  of  gears  consisting  of  a  brass 
pinion  on  a  motor  shaft  driving  a  cast-iron  cut  gear  on  a  counter- 
shaft, which,  in  turn,  drove  the  main  gear  through  another 
pinion.  A  heavy  flywheel  was  mounted  on  the  countershaft  and 
the  backlash  caused  the  repeated  stripping  of  the  teeth  of  the 
cast-iron  gear.  The  brass  pinion  also  rapidly  lost  its  shape  and 
considerable  trouble  was  experienced.  The  idea  of  a  cloth  pinion 
was  then  conceived  and  applied,  and  a  pinion  of  this  material  has 
now  been  in  constant  operation  for  several  years.  It  requires 
no  attention,  runs  without  noise,  and  shows,  as  yet,  no  appreci- 
able signs  of  wear. 

Materials  for  Racks.  —  Racks  of  large  size,  such  as  those 
used  for  driving  the  platens  of  metal  planers,  are  made  of 
iron  or  steel  castings.  Smaller  ones  are  made  from  bar  steel 
stock,  either  machine  steel  finished  all  over,  or  cold-rolled  steel. 
The  latter  material  does  not  require  other  machining  than  the 
cutting  of  the  teeth,  being  accurately  finished  to  certain  con- 
venient sizes  in  the  process  of  rolling.  The  cutting  of  the  teeth 
causes  the  stock  to  bend,  however,  necessitating  a  straightening 
operation. 

The  principle  of  a  device  used  for  the  purpose  of  straightening 
racks  made  from  cold-rolled  steel  is  shown  in  Fig.  i.  This  de- 
vice was  designed  at  the  R.  K.  LeBlond  Machine  Tool  Co.,  Cin- 
cinnati, Ohio.  A  plain  milling  machine  is  used  for  the  purpose. 
An  arbor  is  mounted  in  the  spindle  carrying  ordinarily  three 
gears,  A,  B  and  C,  of  the  pitches  most  commonly  used.  On  the 
table  is  clamped  a  channel  casting  D,  which  is  provided  with  four 
slots  in  each  side  in  which  may  be  placed  the  rollers  E.  The  use 


46  SPUR  GEARING 

of  the  device  is  apparent  upon  inspection.  The  rollers  are 
dropped  into  place  at  such  a  distance  apart  as  best  suits  the 
work  in  hand,  and  are  brought  in  line  with  the  proper  gear  on  the 
arbor,  which  is  then  located  centrally  between  the  two  rollers. 
The  rack  is  now  fed  in  between  the  rollers  and  the  gear,  and  the 
table  is  brought  up  until  enough  pressure  is  exerted  on  the  rack 
to  straighten  it.  The  spindle  is  revolved  slowly  and  the  rack 
feeds  through  the  device  and  is  bent  back  into  shape  by  the 
pressure  between  the  rollers  and  the  pinion.  The  gears  A ,  B  and 
C  are  so  made  that  they  bear  on  the  tops  of  their  teeth  as  well 


Fig.  z.    Straightening  Cold-rolled  Steel  Racks  in  a  Milling  Machine 

as  on  each  side.  This  prevents  stretching  the  racks  when  in 
mesh  with  the  gears.  Were  this  not  so,  the  wedging  action  of 
the  gear  teeth  under  heavy  pressure  would  spread  the  rack  teeth, 
increase  the  pitch,  and,  to  some  degree,  lengthen  the  rack.  This 
device  works  well  for  small  and  medium-sized  racks.  For  larger 
racks,  special  machines  may  be  required,  but  the  principle 
employed  remains  the  same. 

Gears  for  Machine  Tool  Drives.  —  In  a  paper  read  before  the 
American  Society  of  Mechanical  Engineers,  December,  1913, 
Mr.  J.  Parker,  of  the  Brown  &  Sharpe  Mfg.  Co.,  considered  the 


MATERIALS  FOR  GEARS  47 

six  following  questions  relating  to  the  use  of  gears  for  driving 
machine  tools: 

1.  Under  what  conditions  is  it  advisable  to  use  cast-iron  or 
steel  gears  for  machine-tool  drives? 

2.  Are  the  objections  to  cast  iron  on  the  ground  of  wear  or 
breakage? 

3.  What  tooth  pressure  is  safe  for  cast-iron  gears? 

4.  What  grades  of  steel  give  best  results  and  how  should  they 
be  treated? 

5.  How  hard  is  it  advisable  to  make  steel  gears  before  ma- 
chining them? 

6.  Are  they  to  be  hardened  after  machining,  and  if  so,  to 
what  scleroscope  test? 

Conditions  Governing  Use  of  Cast-iron  and  Steel  Gears.  — 
There  are  a  number  of  well-established  gear  conditions  that  are 
common  to  the  majority  of  machine  tools,  which,  if  noted,  may 
prove  somewhat  of  a  guide  in  selecting  the  proper  material  for 
the  gears,  considered  from  the  standpoints  of  economy,  efficiency 
and  durability.  The  conditions  may  be  classified  as  in  the 
accompanying  table.  The  objections  to  cast  iron  cover  both 
wear  and  breakage.  If  the  speed  is  excessive,  say  about  500  feet 
per  minute,  they  are  likely  to  wear  quite  rapidly;  and  on  slow 
speeds  and  heavy  pressure  breakage  will  occur,  unless  they  can 
be  made  of  adequate  size,  as  in  the  case  E,  where  the  back-gears 
are  so  located  in  the  machine  that  it  is  possible  to  employ  large 
diameters,  coarse  pitches  and  wide  faces. 

Safe  Tooth  Pressure  for  Cast-iron  Gears.  —  The  question  of 
tooth  pressures  in  cast-iron  gears  is  somewhat  problematical. 
The  Brown  &  Sharpe  Mfg.  Co.  has  in  successful  operation  a  gear, 
in  the  spindle  drive  of  its  largest  milling  machine,  made  from  a 
hard,  close-grained  cast  iron  having  a  tensile  strength  of  23,000 
pounds  per  square  inch,  which,  when  running  at  the  slowest  speed, 
sustains  a  pressure  on  the  teeth  of  8250  pounds.  It  is  calculated 
that  two  teeth  are  always  hi  contact,  which  gives  4125  pounds 
pressure  per  tooth.  The  area  of  cross-section  of  each  tooth  is  ij 
square  inch,  giving  3300  pounds  per  square  inch.  When  the 
gear  runs  at  the  fastest  speed  the  pressure  is  about  1000  pounds 


48 


SPUR  GEARING 


Gear  Conditions  Common  in  Machine  Tools 


A 

Gears  always  in  mesh, 
the  wear  on  the  teeth 
being  constant. 

(a)  Slow  speeds,  light 
duty 
(6)  Slow  speeds,  heavy 
duty 
(c)  Fast  speeds,   light 
duty 
(d)  Fast  speeds,  heavy 
duty. 

Material 

Cast  iron 
Machine  steel 
Machine  steel 

Machine  steel, 
casehard- 
ened. 

B 

Gears  in  sets  that  are 
removable  and  inter- 
changeable with  each 
other,  distributing 
the  wear  over  a  num- 
ber of  gears. 

These  are  change  gears 
used  in  thread  cut- 
ting on  lathes,  spiral 
cutting  on  milling 
machines,  indexing 
on  automatic  gear 
cutters  and  feed  and 
speed  change  gears; 
speeds  and  pressures 
are  generally  moder- 
ate. 

Cast  iron,  ex- 
cepting the 
smallest, 
which  may 
be  of  steel. 

C 

Gears  in  sets  that  are 
non-removable  and 
partially  inter- 
changeable,  distrib- 
uting the  wear  over 
a  number  of  gears. 
Changes  made  while 
gears  are  in  motion.* 

Used  as  quick-change 
feed  gears  —  changes 
made  by  levers; 
speeds  and  pressures 
moderate. 

Machine  steel, 
casehard- 
ened. 

D 

Gears  in  sets,  that  are 
non-removable  and 
partially  inter- 
changeable, distrib- 
uting the  wear  over 
a  number  of  gears. 
Changes  made  when 
gears  are  at  rest.f 

Used  as  quick-change 
speed  gears  —  changes 
made  by  levers;  high 
speeds  and  heavy 
pressure. 

Machine  steel, 
casehard- 
ened. 

E 

Gears  that  are  em- 
ployed only  part  of 
the  time  the  machine 
is  working,  and  are 
engaged  and  disen- 
gaged when  the  ma- 
chine is  stopped. 

This  condition  applies 
to  back-gears  for  the 
spindle  drive.  Gears 
are  made  large  diam- 
eter, coarse  pitch 
and  wide  face;  speeds 
moderate,  and  heavy 
pressure. 

Hard,     close- 
grained  cast 
iron. 

*  If  the  changes  were  made  when  the  machine  was  at  rest,  the  gears  would  not  require  hard- 
ening, but  custom  demands  that  changes  be  made  while  the  machine  is  running. 

t  Although  the  changes  are  supposed  to  be  made  when  gears  are  at  rest,  careless  workmen 
will  violate  this  rule,  with  the  possibility  of  breaking  the  engaging  gears.  Some  makers  use  an 
alloy  steel  in  their  spindle  train  to  prevent  breakage,  but  a  better  way  is  to  provide  means  where- 
by it  is  necessary  to  stop  the  machine  before  throwing  in  the  gears.  This  applies  to  the  tumbler 
type  of  change  gearing. 


MATERIALS  FOR  GEARS  49 

per  square  inch.  It  is  not  known  whether  the  pressure  could  be 
increased  to  any  considerable  extent,  but  the  gear  has  been  over- 
loaded to  at  least  30  per  cent  without  injuring  it;  this  was  when 
testing  out  the  machine  and  the  overload  was  of  short  duration. 
It  might  be  said  that  this  gear  is  not  subjected  to  any  sudden 
shock;  if  it  were,  the  tooth  pressure  would  have  to  be  less. 

Grades  of  Steel  that  give  Best  Results.  —  For  gears  that  are 
of  small  proportions  and  yet  are  subjected  to  heavy  duty,  it  has 
been  found  that  in  cases  where  the  more  common  steels  have 
failed,  excellent  results  have  been  obtained  by  using  a  5  per 
cent  nickel  steel.  This  steel  casehardens  with  a  very  hard  sur- 
face and  still  has  a  strong  and  tough  core,  making  it  an  ideal  steel 
to  use  where  the  pressure  is  heavy  or  the  gear  is  subjected  to 
shock.  Experience  shows  that  drop-forgings  are  more  uniform 
in  texture  than  bar  stock. 

Heat-treatment  of  Steel  for  Gears.  —  The  5  per  cent  nickel 
steel  referred  to  in  the  preceding  paragraph  is  given  an  oil  treat- 
ment and  is  also  annealed  before  machining.  The  oil  treatment 
is  as  follows:  Heat  to  1550  degrees  F.  and  quench  in  oil.  To 
anneal,  reheat  to  1350  degrees  F.  and  cool  very  slowly.  The 
steel  is  then  ready  to  machine.  After  machining,  it  is  carbonized 
as  follows:  Pack  in  any  good  carbonizing  material  and  cover  very 
carefully  to  exclude  air;  place  in  furnace  and  heat  to  1700 
degrees  F.,  and  hold  long  enough  to  get  the  desired  depth  of 
casing.  Care  should  be  taken  to  have  the  steel  heated  entirely 
through.  Ordinarily  from  three  to  four  hours  is  sufficient. 
Then  remove  the  steel  from  the  furnace  and  cool  in  the  boxes; 
next  remove  from  the  boxes  and  place  in  a  furnace  or  bath; 
reheat  to  1550  degrees  F.  and  quench  in  oil.  Again  reheat  to 
about  1380  degrees  F.  and  quench  in  oil  or  water,  according  to 
the  size  and  shape  of  gear.  If  the  gear  is  of  generous  dimensions 
and  free  from  sharp  corners,  water  is  preferable.  Small  slender 
gears  are  quenched  in  oil,  on  account  of  the  liability  of  cracking 
if  water  is  used.  For  ordinary  gears  the  scleroscope  test  should 
show  80  to  85  points  of  hardness.  If  the  gears  are  used  as  clash 
gears  they  should  be  drawn  to  475  degrees  F.,  or  about  70  to 
75  points  of  hardness,  by  scleroscope  test,  to  avoid  chipping. 


50  SPUR   GEARING 

Hardness  Advisable  for  Steel  Gears  before  Machining.  — 

The  kinds  of  steels  generally  used  for  gears  are  of  such  a  nature 
that  they  do  not  call  for  heat-treatment  before  machining,  but 
where  extra  toughness  is  required  to  withstand  torsion  and 
bending  strains,  3!  per  cent  nickel  steel  is  satisfactory  and  is 
heat-treated  as  follows,  after  being  rough-machined:  Place  in 
an  open  furnace  or  bath,  heat  to  1500  degrees  F.  and  quench  in 
oil.  It  is  advisable  to  experiment  with  a  small  quantity  in  each 
batch,  before  subjecting  a  whole  lot  to  the  drawing- out  heat, 
which  should  begin  at  about  700  degrees  F.  If  the  scleroscope 
registers  between  50  and  58,  the  correct  hardness  has  been 
obtained;  if  higher  than  58,  the  parts  should  be  reheated  to  a 
higher  temperature  than  before;  if  lower  than  50,  the  parts  must 
be  rehardened.  After  this  treatment,  the  pieces  are  finish- 
machined.  No  further  hardening  is  necessary.  When  machin- 
ing, slow  speeds  and  feeds  must  be  used. 

Hardening  after  Machining,  and  the  Scleroscope  Test.  — 
Practically  all  alloy  and  all  low-carbon  steels  are  hardened  after 
machining  and  finished  by  grinding  after  hardening.  About 
o.oio  inch  on  the  diameter  is  left  for  this  operation.  All  gears 
should  run  true,  and  to  obtain  this  result  not  only  are  the  holes 
ground  true  with  the  pitch  circle,  but  the  hubs  are  ground  on 
their  faces  so  that  they  will  set  square  with  their  shafts  when 
tightened  up  by  nuts.  The  scleroscope  test  for  0.30  to  0.35 
per  cent  carbon  machine  steel  is  anywhere  from  80  to  90,  and  for 
5  per  cent  nickel  steel  for  ordinary  gears,  80  to  85,  and  for  clash 
gears,  70  to  75.  All  steels  are  tested  by  the  file  in  addition  to 
the  scleroscope.  The  file  test  by  an  expert  is  very  reliable  and 
some  feel  that  possibly  more  confidence  can  be  placed  on  his 
judgment  than  on  any  testing  instrument. 

The  above  notes  apply  to  spur  and  bevel  gears.  For  worm 
and  worm-wheel  drives,  the  worm  should  be  made  of  machine 
steel,  casehardened,  and  the  wheel  of  a  hard  bronze.  Both  should 
run  in  a  bath  of  oil,  especially  if  working  under  high  speed  and 
heavy  duty.  Spiral  gears  should  be  used  only  where  the  duty 
is  light.  The  material  should  be  the  same  as  for  a  worm  and 
worm-wheel,  and  they  should  also  run  in  oil  to  avoid  cutting. 


CHAPTER  III 
STRENGTH  AND  DURABILITY  OF  SPUR  GEARING 

Strength  of  Gear  Teeth.  —  There  is  a  great  deal  of  discrepancy 
between  the  various  rules  published  by  different  authorities, 
largely  due  to  the  many  varying  conditions  met  with  in  gearing, 
these  conditions  depending  upon  the  materials,  the  methods  of 
producing  the  gear  teeth,  the  conditions  of  service,  etc.  It  might 
be  well  said  that  no  rule  for  the  strength  of  gear  teeth  is  com- 
plete unless  it  states  definitely  for  what  class  of  gears  it  is  in- 
tended. For  cut  gears  the  Lewis  formula,  given  in  the  following, 
is  accepted  as  standard.  For  cast  gears  it  would  seem  that  the 
best  method  would  be  to  use  the  Lewis  formula  so  modified  as  to 
give  a  factor  of  safety  two  or  three  times  that  required  by  this 
formula,  depending  upon  the  severity  of  the  conditions  in  each 
case.  A  general  rule  for  cast  gears,  used  under  so  many  varying 
conditions,  could  hardly  be  satisfactorily  devised.  The  designer's 
judgment  must  be  depended  upon  to  a  considerable  extent  in  the 
selection  of  the  suitable  factor  of  safety  which  will  be  required  in 
excess  of  that  required  for  cut  gears. 

As  an  indication  of  the  uncertainty  of  the  rules  given  by 
various  authorities,  Mr.  Lewis  mentions  in  his  paper  on  the 
"Strength  of  Gear  Teeth,"  read  before  the  Engineers'  Club  of 
Philadelphia,  October  15,  1892,  and  published  in  the  proceedings 
of  the  club,  January,  1893,  tnat  Mr.  J.  H.  Cooper  in  making  an 
investigation  found  that  there  were  not  less  than  forty-eight 
well-established  rules  in  existence  for  the  strength  and  capacity 
of  gear  teeth,  these  rules  being  sanctioned  by  twenty-four  differ- 
ent authorities  and  giving  gear  teeth  varying  500  per  cent  in 
ultimate  strength. 

The  Lewis  Formula.  —  The  accompanying  tables  of  "  Rules 
and  Formulas  for  the  Strength  of  Gear  Teeth,"  "Factors  for 
Calculating  the  Strength  of  Gear  Teeth"  and  "Working  Stresses 

Si 


SPUR  GEARING 


used  in  the  Lewis  Formula  for  the  Strength  of  Gear  Teeth"  make 
it  possible  to  quickly  calculate  the  strength  of  spur  gears.  The 
formulas  and  factors  given  are  based  on  the  use  of  the  diametral 
pitch,  and  the  constants  Y  given  in  the  factor  table  are  valid  only 
when  the  diametral  pitch  is  used.  If  the  circular  pitch  is  given, 
it  should  be  transformed  into  diametral  pitch  by  dividing  3.1416 
by  the  circular  pitch.  By  means  of  the  formulas  given,  the 

Working  Stresses  Used  in  the  Lewis  Formula  for  the  Strength  of 
Gear  Teeth 


Safe  Working  Unit  Stress  =  S,  in  Pounds  per  Square  Inch 

8 

Q 

li 

£ 

Cast  Iron 

Phosphor  Bronze 

Steel 

>..g 

1 

Ordinary 

High-grade 

Ordinary 

High-grade 

Ordinary 

High-grade 

1 

CO 

Workman- 

Workman- 

Workman- 

Workman- 

Workman- 

Workman- 

ship 

ship 

ship 

ship 

ship 

ship 

o 

1  .000 

6000 

8000 

9000 

I2,OOO 

I5,OOO 

2O,OOO 

IOO 

0.857 

5I5° 

6850 

7700 

10,300 

I2,8oo 

17,100 

2OO 

0.750 

4500 

6000 

6750 

9,OOO 

II,2OO 

15,000 

3OO 

0.666 

4000 

5350 

6000 

8,000 

IO,OOO 

13,300 

450 

o.57i 

3400 

4550 

5150 

6,850 

8,550 

11,400 

600 

0.500 

3000 

4OOO 

4500 

6,000 

7,500 

10,000 

900 

0.400 

2400 

3200 

3600 

4,800 

6,000 

8,000 

1200 

0-333 

2OOO 

2650 

3000 

4,000 

5,000 

6,650 

1800 

0.250 

1500 

2OOO 

2250 

3,000 

3,750 

5,ooo 

24OO 

0.200 

I2OO 

I6OO 

1800 

2,400 

3,ooo 

4,000 

horsepower  which  can  be  transmitted  by  a  gear  of  a  given  pitch 
diameter  and  diametral  pitch,  running  at  a  given  number  of 
revolutions  per  minute,  can  be  found  by  using  Formulas  (i)  to 
(4)  in  the  order  given.  The  allowable  static  unit  stress  for  the 
material  in  the  gear  is  selected  from  the  first  line  (velocity  =  o) 
in  the  table  of  working  stresses;  the  stress  at  any  given  velocity 
may  also  be  found  directly  from  the  table. 

The  utility  of  the  Lewis  formula  is  due  to  its  simple  form,  to 
the  fact  that  it  takes  into  account  a  greater  number  of  factors 
than  does  any  other,  and  to  the  fact  that  the  effect  of  each  of 
these  factors  is  rationally  expressed  in  the  formula. 

Example  of  Application  of  Formula.  —  As  an  example  of  the 
application  of  the  Lewis  formula  to  an  actual  problem,  assume 


STRENGTH  AND  DURABILITY 


53 


that  it  is  required  to  find  the  horsepower  which  it  is  permissible 
to  transmit  by  a  spur  gear  having  1 5-inch  pitch  diameter,  4 
diametral  pitch,  making  100  revolutions  per  minute,  and  having 
a  width  of  face  i£  inch,  the  teeth  being  cut  according  to  the  14!- 
degree  involute  system.  The  gear  is  made  of  steel  and  the  allow- 

Rules  and  Formulas  for  the  Strength  of  Gear  Teeth 

(Based  on  the  Lewis  Formula) 


D  =  pitch  diameter  of  gear  in  ins.  ;     A  =  width  of  face  in  inches; 
R  =  revolutions  per  minute  ;                Y=  outline  factor  (see  table)  ; 
V=  velocity  in  ft.   per  min.   at     P  =  diametral    pitch     (if    circular 
pitch  diameter;                                    pitch  is  given,  divide  3.1416 
Ss  =  allowable   static  unit  stress                by  circular  pitch  to  obtain 
for  material;                                          diametral  pitch); 
S=  allowable  unit  stress  for  ma-     W=  maximum  safe  tangential  load 
terial  at  given  velocity;                     in  pounds  at  pitch  diameter; 
H.P.=maximum  safe  horsepower. 
Use  rules  and  Formulas  (i)  to  (4)  in  the  order  given. 

No. 

To  Find 

Rule 

Formula 

« 

Velocity  in  feet 
per  min.  at  the 
pitch  diameter. 

Multiply  the  product  of  the 
diameter  in  inches  and  the 
number  of  revolutions  per 
minute,  by  0.262. 

7=0.262  DR 

o 

Allowable  unit 
stress  at  given 
velocity. 

Multiply  the  allowable 
static  stress  by  600  and  divide 
the  result  by  the  velocity  in 
feet  per  minute  plus  600. 

0           0    y           600 

~^X6oo+F 

c) 

Maximum  safe 
tangential  load  at 
pitch  diameter. 

Multiply  together  the  allow- 
able stress  for  the  given  ve- 
locity, the  width  of  face,  and 
the     tooth     outline     factor; 
divide  the  result  by  the  diam- 
etral pitch. 

SAY 

P 

<?1 

Maximum  safe 
horsepower.  v 

Multiply  the  safe  load  at 
the  pitch  line  by  the  velocity 
in  feet  per  minute,  and  divide 
the  result  by  33,000. 

II  T        WV 

33,000 

able  static  unit  stress  for  the  material  may,  therefore,  be  assumed 
to  be  15,000  pounds  per  square  inch.  First  insert  the  values  of 
the  revolutions  per  minute  and  the  pitch  diameter  in  Formula  (i) 
and  thus  find  the  velocity  in  feet  per  minute  at  the  pitch  diameter. 
V  =  0.262  X  15  X  ioo  =  393  feet  per  minute. 


54 


SPUR  GEARING 


This  velocity,  together  with  the  allowable  static  unit  stress,  is 
then  inserted  in  Formula  (2),  and  the  allowable  unit  stress  at  the 
given  velocity  is  found. 

600 


S  =  15,000  X 


600  +  393 


=  9000  pounds,  approx. 


This  unit  stress  is  now  inserted  in  Formula  (3)  together  with 
the  width  of  face,  the  outline  factor  Y  (which  is  found  from  the 
table  to  be  0.358  for  60  teeth)  and  the  diametral  pitch;  in  this 
way,  the  maximum  safe  tangential  load  W  is  found. 

Factors  for  Calculating  Strength  of  Gear  Teeth  (  P/a* 


Outline  Factor 

Outline  Factor 

Outline  Factor 

=  Y 

-  Y 

=  Y 

No.  of 
Teeth 

14^°  In- 

No. of 
Teeth 

14^°  In- 

No. of 
Teeth 

I4H°  In- 

volute 

20°  In- 

volute 

20°  In- 

volute 

20°  In- 

and Cy- 

volute 

and  Cy- 

volute 

and  Cy- 

volute 

cloidal 

cloidal 

cloidal 

12 

O.2IO 

0.245 

20 

0.283 

0.320 

43 

0.346 

0.396 

13 

O.22O 

0.261 

21 

0.289 

0.327 

SO 

0.352 

0.408 

14 

O.226 

0.276 

23 

0.295 

0-333 

60 

0.358 

0.421 

is 

0.236 

0.289 

25 

0.305 

0-339 

75 

0.364 

0-434 

16 

0.242 

0.295 

27 

0.314 

0-349 

100 

0.371 

0.446 

17 

0.251 

0.302 

30 

0.320 

0.358 

150 

0-377 

0-459 

18 

19 

0.26l 
0.273 

0.308 
0.3U 

34 
38 

0.327 
0.336 

0.371 
0.383 

300 
Rack 

0-383 
0.390 

0.471 
0.484 

jj,        0000  X  1.5  X  0.358 

W  =  2 J :2^—  =  1210  pounds,  approx. 

4 

Finally,  by  inserting  the  value  of  W  just  found  and  the  value 
V  found  from  Formula  (i)  in  Formula  (4),  we  determine  the 
maximum  safe  horsepower  which  can  be  transmitted  by  the  gear. 


H  p  =  1210  X  393  = 
33,000 


14.4. 


General  Remarks  Relating  to  the  Use  of  the  Tables  and 
Formulas.  —  It  will  be  noted  that  two  columns  of  values  are 
given  for  each  material  in  the  table  "Working  Stresses  used  in  the 
Lewis  Formula  for  the  Strength  of  Gear  Teeth  " ;  as  explained,  the 


STRENGTH  AND  DURABILITY  5£ 

first  set  of  values  may  be  used  for  workmanship  of  ordinary 
grade,  while  the  other  is  permissible  with  a  higher  grade.  The 
second  column,  of  strength  factors,  may  be  used  for  finding  the 
allowable  working  fiber  stress  to  use  for  any  given  speed,  when 
the  safe  static  stress  is  known;  to  find  the  fiber  stress,  multiply 
the  safe  static  stress  by  the  strength  factor.  Formula  (2)  in 
"Rules  and  Formulas  for  the  Strength  of  Gear  Teeth"  may  be 
used  for  the  same  purpose,  giving  results  closely  approximating 
the  values  in  the  table. 

The  variable  factor  introduced  into  the  problem  by  the  vary- 
ing shape  of  teeth  of  the  same  pitch,  in  gears  of  different  numbers 
of  teeth,  is  taken  care  of  by  introducing  in  the  formula  the  outline 
factors  Y  given  in  the  table  "Factors  for  Calculating  Strength  of 
Gear  Teeth."  These  factors  are  given  for  that  arrangement  of 
the  formula  which  applies  to  diametral  pitches. 

The  formulas  in  the  chart  take  no  account  of  any  such  limita- 
tion of  the  width  of  face  for  a  gear  of  given  pitch  as  obtains  in 
practice.  The  strength  is  made  to  increase  directly  with  the 
width  of  face,  without  limit.  In  practice,  if  the  face  is  too  long 
in  proportion  to  the  size  of  the  tooth,  we  cannot  be  sure  that 
each  tooth  of  the  gear  has  a  full  bearing  over  its  whole  length  on 
its  mating  tooth  in  the  other  gear.  The  shafts  on  which  the  two 
are  mounted  may  not  be  parallel,  or,  if  originally  parallel,  they 
may  deflect  under  the  strain  of  the  load  transmitted.  For 
reasons  like  this,  in  ordinary  commercial  work,  the  width  of  face 
may  be  considered  as  well  proportioned  when  it  equals,  in  inches, 
8.75  divided  by  the  diametral  pitch.  As  a  formula,  this  gives 


In  cases  where  accurate  workmanship  can  be  depended  on, 
there  is  a  gain  in  using  teeth  of  wider  face  and  finer  pitch  than 
would  be  allowed  by  Formulas  (i)  to  (4).  There  is  a  gain  in 
efficiency,  smoothness  of  action,  and  noiselessness,  especially  at 
high  speeds.  In  fact,  the  width  of  face  of  a  gear  may  well  be 
made  to  depend  in  part  on  the  speed  at  which  it  is  run,  as  well  as 
on  the  pitch,  it  being  taken  for  granted,  of  course,  that  the  pitch 
and  width  are  such  as  to  give  the  required  strength. 


56  SPUR  GEARING 

Relation  between  Width  of  Face  and  Diametral  Pitch.  —  The 

rules  and  formulas  for  the  strength  of  gear  teeth  can  be  used  for 
finding  the  pitch  and  width  of  gear  teeth  for  transmitting  a  given 
horsepower,  if  used  in  combination  with  a  formula  giving  the 
best  relation  between  pitch  and  width  of  face.  An  accepted 
rule  for  this  relation  is  as  follows:  To  find  a  well-proportioned 
width  of  face  for  carefully  made  gearing,  multiply  the  square 
root  of  the  pitch  line  velocity  in  feet  per  minute  by  0.15,  add  9 
to  the  product,  and  divide  the  result  by  the  diametral  pitch;  or, 
if  A  =  width  of  face  in  inches;  V  =  pitch  line  velocity  in  feet 
per  minute;  and  P  =  diametral  pitch,  then: 

A  _2i 


Example:  —  What  should  be  the  pitch  and  the  width  of  face 
of  a  steel  pinion,  4  inches  in  pitch  diameter,  the  teeth  shaped 
according  to  the  i/^-degree  involute  system,  running  750  revolu- 
tions per  minute,  and  transmitting  10  horsepower?  The  work- 
manship is  high  grade,  and  the  width  of  the  face  is  to  be  pro- 
portioned according  to  the  rule  and  formula  given  immediately 
above. 

The  velocity  at  the  pitch  line  =  0.262  DR  =  0.262  X  4  X 
750  =  786  feet  per  minute.  (See  Formula  (i)  in  the  table  for 
strength  of  spur  gears.)  The  allowable  running  stress  for  a 
static  stress  of  20,000  pounds  per  square  inch  is  found  by  For- 
mula (2)  : 


S  =  20,000  X  -  —  =  8660  pounds  per  square  inch. 

ooo  +  780 

The  load  at  the  pitch  line  is  equal  to 

10  X  33,000  X  12 

-  =  420  pounds. 
TT  X  4  X  750 

Assume  5  diametral  pitch  as  a  trial  pitch  for  the  teeth,  then  the 
number  of  teeth  equals  5  X  4  =  20.    Transposing  Formula  (3)  : 

w      SAY      .  WP 

W  =  --  ,  gives  A= 


STRENGTH  AND   DURABILITY  57 

Apply  this  transposed  formula  and  find  a  trial  width  of  face 
(factor  F  is  given  in  the  table  of  "  Factors  for  Calculating 
Strength  of  Gear  Teeth"): 


5  —  _ 


approximately. 


8660  X  0.283 
For  5  pitch,  however,  according  to  the  formula 

A  _  °-I5 


P 

the  width  of  the  face  should  be 


0.1  9         ,    •     , 

A  =  -  2  =  2.64  inch. 

Thus,  the  pitch  is  evidently  too  coarse.  Repeated  trials  show 
that  with  9  diametral  pitch  the  results  from  the  two  formulas  for 
width  of  face  agree  fairly  well.  Thus: 

Number  of  teeth  =  9  X  4  =  36. 

420  X  o  .1 

A  —-  —  *  —  =  1.32  inch. 

8660  X  0.332 

A       0.15  A/786  +  9  .    , 

A  =  —  J  —  L—   -1-2  =  1.47  inch. 

9 

Hence,  9  diametral  pitch  and  if  -inch  width  of  face  are  the 
dimensions  to  which  the  gear  ought  to  be  made. 

Derivation  of  the  Lewis  Formula.  —  The  factor  Y  in  the  Lewis 
formula  is  derived  as  follows:.  In  Fig.  i  a  i4^-degree  involute 
tooth  is  shown  in  outline.  The  pressure  F  transmitted  to  the 
gear  tooth  at  an  angle  of  14  J  degrees  to  a  tangent  to  the  pitch 
circle  may  be  considered  as  having  two  components,  one  of  which 
is  radial  and  tending  to  crush  the  tooth,  and  one  tangential, 
tending  to  break  off  the  tooth.  The  only  component  we  need 
to  consider  in  connection  with  the  strength  of  gear  teeth  is  the 
tangential  component  W.  This  component,  as  shown  in  Fig.  i, 
is  not  considered  as  acting  upon  the  tooth  at  its  top,  but  at  the 
point  B  where  the  line  representing  the  pressure  F  intersects  the 
center  line  of  the  tooth.  The  line  of  pressure,  of  course,  is  con- 
sidered as  passing  through  the  corner  of  the  tooth  as  indicated. 


SPUR  GEARING 


Any  parabola  having  the  line  BE  for  an  axis  and  passing 
through  the  point  B,  so  that  the  line  representing  force  W  is  a 
tangent  to  the  parabola  at  this  point,  encloses  a  beam  of  uniform 
strength.  By  graphical  construction  it  can  now  be  determined 
in  each  case  where  the  point  of  tangency  between  the  parabola 
and  the  contour  of  the  tooth  form  occurs.  The  point  of  tangency 
determines  the  weakest  section  of  the  tooth,  as  shown  at  CD. 
This  weakest  section  is  not  necessarily  at  the  very  root  of  the 
tooth,  but  may  —  particularly  in  gears  with  a  small  number  of 
teeth  —  be  located  a  considerable  distance  above  the  root  of  the 
tooth;  hence  the  distance  L,  the  length  of  the  beam,  is  measured 
not  to  the  bottom  of  the  tooth,  but  to  the  line  CD. 


Machinery,  N.Y. 


Fig.  i.    Diagram  used  in  the  Derivation  of  the  Lewis  Formula 

In  order  to  determine  the  value  Y  in  the  Lewis  formula, 
draw  the  lines  BC  and  CE,  the  latter  being  at  right  angles  to 
BC  and  intersecting  the  center  line  of  the  tooth  at  E.    In  the 
following  formulas : 
S  =  the  fiber  stress; 
A  —  the  width  of  the  face  of  the  gear; 
P  —  circular  pitch  of  gear; 

L,  M,  T  and  W  denote  quantities  as  indicated  in  the  illus- 
tration. 
Then  we  have: 

TTT  T  t^A  J-  -rfT  «J-^1  -/ 

WL  =  -  —  ,  or  W  =  — - 
6  6L 


STRENGTH  AND  DURABILITY 
But,  by  similar  triangles, 

M=^L 


59 


W  =SA  x-M  =  SAP- 


Hence 


where  — — •  =  F,  the  factor  in  the  Lewis  formula,  for  circular 
pitch. 


Machinery 


Fig.  2.    Method  of  applying  Load  to  Tooth  for  Testing  Strength 

Tests  for  Strength  of  Gear  Teeth.  —  In  a  paper  read  before 
the  National  Machine  Tool  Builders'  Association,  October,  1913, 
Mr.  A.  C.  Gleason  gave  some  information  relating  to  tests  of  the 
strength  of  heat-treated  gears.  The  accompanying  table  gives 
the  average  results  obtained  from  a  considerable  number  of  test 
pieces  subjected  to  the  different  heat-treatments  specified.  The 
test  pieces  were  spur  pinions  having  fourteen  teeth  of  six  pitch, 
2\  inches  pitch  diameter,  i-inch  face  width  and  i-inch  bore.  All 
teeth  were  of  standard  depth  and  shape.  During  the  tests  the 
pinions  were  held  in  a  special  chuck  as  shown  in  the  accompany- 


6o 


SPUR  GEARING 


Ultimate  Strength  and  Elastic  Limit  of  Gear  Teeth  for  Different  Steels 
and  Heat-treatments 


No.  of 
Test 

Heat-treatment 

Elastic 
Limit 

Breaking 
Strain 

0.20  per  cent  Carbon,  Open-hearth  Casehardening  Steel 

No.  i. 
No.  2. 

No.  3. 

Soft  

3.500 
8,400 
8,000 

8,200 
8,000 

Casehardened  with  one  tempering  heat  of 
1450°  F.,  90  scleroscope  hardness  

9,000 
9.450 

9.675 
9,800 

Same  drawn  to  400°  F.,  85  scleroscope  hard- 
ness .                        ... 

Casehardened  with  two  tempering  heats  of 
1600°  F.,  and  1450°  F.,  85  scleroscope  hard- 
ness   

Same  drawn  to  400°  F.,  80  scleroscope  hard- 
ness   

i}4  per  cent  Nickel,  0.18  per  cent  Carbon,  Natural  Alloy  Steel 

No.  i. 
No.  2. 

No.  3. 

Soft 

4,000 

9,000 
8,600 
8,750 
8,400 

Casehardened    with    one    tempering    heat, 
92  hardness 

10,150 
10,450 
IO,6oo 
10,750 

Same  drawn  to  400°  F.,  87  hardness 

Casehardened  with  two  tempering  heats.  .  . 
Same  drawn  to  400°  F.,  82  hardness  

3H  per  cent  Nickel,  0.13  per  cent  Carbon,  Open-hearth  Nickel  Alloy 

No.  i. 
No.  2. 

No.  3. 

Soft  

4,000 
9.790 

9.5oo 
9*250 

Casehardened  with  one  tempering  heat  of 
1350°  F.,  90-95  hardness  

11,400 

11,650 
ii,95° 

Casehardened  with  two  tempering  heats  of 
1550°  F.,  and  1350°  P.,  90-95  hardness.  .  .  . 
Same  drawn  to  400°  F.,  85  hardness  

5  per  cent  Nickel,  0.15  per  cent  Carbon.  Open-hearth  Alloy  Steel 

No.  i. 

No.  2. 

No.  3. 

Soft  

4.5oo 

13,000 
12,700 

13,000 
12,800 

Casehardened  with  one  tempering  heat  of 
1350°  F.,  90  hardness.  ... 

13,880 
14,800 

14,100 
14,850 

Same  drawn  to  400°  F.,  85  hardness   . 

Casehardened  with  two  tempering  heats  of 
1550°  F.,  and  1350°  F.,  90  hardness  

Same  drawn  to  400°  F.,  85  hardness  

(A)  Chrome-nickel  Tempering  Steel 

No.  i. 
No.  2. 

Quenching  heat  1425°  F.,  drawing  tempera- 
ture 475°  F.,  65—70  hardness. 

22,450* 

Soft  .  . 

<(,OOO 

(B)  Chrome-nickel  Tempering  Steel 

No.  i. 
No.  2. 
No.  3. 

Quenching  heat  1480°  F.,  drawing  tempera- 
ture, 525°  F.,  75—78  hardness 

17,640* 
19,440* 

Quenching  heat,  1480°  F.,  drawing  temper- 
ature, 850°  F.,  60-65  hardness  

Soft  

5,200 

*  No  set  before  breaking. 


STRENGTH  AND   DURABILITY  6 1 

ing  illustrations,  the  load  being  applied  to  the  edge  of  the  teeth 
as  indicated  in  Fig.  2.  The  initial  displacement  or  elastic  limit 
was  determined  by  following  up  the  load  with  a  vernier  tooth 
caliper  which  showed  when  permanent  set  had  occurred  (see 
Fig.  3).  All  carbonizing  was  Jg-  inch  deep,  and  the  furnace 
temperature  varied  from  1450  to  1500  degrees  F.,  for  eight  hours, 
and  was  held  at  1600  degrees  for  from  three  to  five  hours  —  a 
total  of  eleven  to  thirteen  hours. 


Machinery 


Fig.  3.    Gear-tooth  Caliper  used  to  determine  Permanent  Deflection 

It  will  be  noted  that  the  increased  strength,  as  a  result  of 
the  double  heat-treatment  after  carbonizing,  is  comparatively 
slight,  particularly  in  the  nickel-alloy  steels.  This  is  undoubtedly 
due  to  the  long  and  low  carbonizing  heats  and  the  low  hardening 
heats.  The  various  hardened  test  pieces  of  the  "straight- 
carbon"  steel  showed  a  variation  in  strength  of  10  per  cent  above 
and  below  the  records  given.  This  stock  was  selected  especially 
for  casehardening.  The  ordinary  run  of  machine  steel  of  the 
same  carbon  content  will  vary,  in  some  cases,  twice  as  much  as 


62  SPUR  GEARING 

this.  The  same  may  be  said  of  the  so-called  "natural  nickel- 
alloy  steels; "  they  vary  as  much  as  the  straight-carbon  steels. 
The  higher  grades  of  alloy  steels  do  not  vary  more  than  10  per 
cent  either  way,  and  the  average  is  closer. 

Rawhide  Gearing.  —  When  rawhide  gears  were  first  intro- 
duced the  rawhide  pinion  occupied  a  peculiar  position,  in  that  it 
was  required  to  demonstrate  its  qualities  by  a  satisfactory  per- 
formance of  service  for  which  cast-iron,  steel  or  bronze  pinions 
had  originally  been  designed.  In  the  electric  railway  field,  for 
instance,  rawhide  pinions  have  thus  performed  the  work  origi- 
nally intended  for  steel  and  bronze  gears,  and  successfully  com- 
peted with  the  bronze  on  the  basis  of  mileage  cost.  Of  course, 
rawhide  is  not  quite  as  durable  as  steel,  but  it  is  often  used  in 
preference  to  steel,  because  it  will  run  more  quietly  and  the  life 
of  certain  other  motor  parts  is  increased  because  of  the  decrease 
in  vibration.  This  applied  especially  to  the  motors  formerly 
used  having  high  gear  velocities  and  comparatively  low  tooth 
loads.  Mr.  W.  H.  Diefendorf ,  chief  engineer  of  the  New  Process 
Gear  Corporation,  Syracuse,  N.  Y.,  mentions  in  one  of  his  pub- 
lished statements  that  actual  performances  have  thoroughly 
demonstrated  that,  at  a  peripheral  velocity  of  from  1700  to  2000 
feet  per  minute  and  more,  rawhide  pinions  can  be  advantageously 
used  in  place  of  cast-iron  and  bronze  pinions,  and,  occasionally, 
even  in  place  of  steel  pinions.  The  fact  that  this  has  been  done 
has  naturally  led  to  the  belief  that  rawhide  pinions  are  equal  to 
metal  pinions  under  all  conditions.  At  speeds  under  1700  feet, 
however,  owing  to  the  increased  tooth  loads  permissible,  by 
reason  of  the  reduction  of  shock,  it  will  be  found  necessary  to  use 
rawhide  pinions  of  either  larger  diameters  or  of  longer  face,  or 
both,  than  would  be  used  for  metal  pinions.  The  meshing  gears 
must,  of  course,  be  proportioned  accordingly. 

Allowable  Load  Per  Inch  of  Face  of  Rawhide  Gears.  —  The 
safe  working  load  for  a  rawhide  pinion  of  the  highest  grade  is  150 
pounds  per  inch  width  of  face  for  one-inch  circular  pitch.  Other 
pitches  vary  in  direct  proportion,  except  that  the  maximum  load 
must  not  exceed  250  pounds  per  inch  width  of  face.  The  reason 
why  the  capacity  does  not  increase  indefinitely  with  the  increase 


STRENGTH  AND  DURABILITY  63 

of  circular  pitch,  as  is  within  reasonable  limits  the  case  with 
metal  gears,  is  that  at  a  point  not  far  above  250  pounds  load 
per  inch  width  of  face  the  surface  of  rawhide  begins  to  compress, 
thereby  permitting  a  distortion  of  the  tooth  curve,  resulting  in 
friction  and  causing  undue  heating.  A  temperature  of  225  de- 
grees F.  is  more  than  a  rawhide  pinion  should  be  subjected  to. 
In  exceptional  cases,  rawhide  pinions  have  worked  successfully 
for  two  or  three  months  continuously  at  a  load  of  from  350  to  450 
pounds  per  inch  width  of  face,  but  when  removed  from  service 
the  sides  of  the  teeth  were  hardened  to  a  depth  of  J  inch  or  more, 
and  the  material  resembled  hard  glue  or  rosin  and  had  lost  all 
its  elasticity. 

Formula  for  Power  Transmitted  by  Rawhide  Gears.  —  In  the 
following  formula,  given  by  the  New  Process  Gear  Corporation, 

P  =  circular  pitch  in  inches; 
F  —  width  of  face  of  gear  in  inches; 
D  =  pitch  diameter; 
N  —  number  of  revolutions  per  minute; 
H.  P.  =  horsepower  transmitted. 


Then,        H.P  ^ 

850 

In  using  this  formula,  the  circular  pitch  should  not  be  given  a 
higher  value  than  1.65  inch,  in  order  to  limit  the  total  load  to  150 
pounds  per  inch  width  of  face  for  one-inch  circular  pitch,  as  pre- 
viously stated.  Of  course,  rawhide  gears  of  a  greater  circular 
pitch  than  1.65  inches  may  be  used  to  advantage,  but  when  cal- 
culating the  horsepower  transmitted  by  these  gears  the  value 
for  P  in  the  formula  should  never  be  given  as  larger  than  1.65. 

The  denominator  850  is  used  for  the  highest  grade  of  rawhide 
gears.  When  lower  grades  of  material  are  used,  the  factor  in 
the  denominator  should  be  increased  from  850  to  about  1000.  It 
will  be  noted  that  in  the  formula  no  allowance  is  made  for  periph- 
eral velocity  and  number  of  teeth.  This  is  because  rawhide 
pinions  are  intended  primarily  for  high-speed  service,  where  an 
all-metal  drive  would  be  noisy.  As  a  result,  the  peripheral 
velocities  are  high  and  the  cushioning  effect  of  the  rawhide  teeth 


64  SPUR  GEARING 

compensates  for  the  usual  factor  of  shock.  The  number  of  teeth 
in  rawhide  pinions  is  made  from  fifteen  to  eighteen  wherever 
possible. 

Example:  —  As  an  example  of  the  use  of  the  horsepower 
formula  given,  find  the  horsepower  transmitted  by  a  rawhide 
pinion  6.37  inches  pitch  diameter,  ij  inch  circular  pitch,  6  inches 
width  of  face  and  running  at  1200  revolutions  per  minute. 

H  p       6.37  X  1.25  X  6  X  1200  = 
850 

Strength  of  Rawhide  Gears  with  Flanges.  —  The  information 
relating  to  the  strength  of  rawhide  gears,  given  in  the  previous 
paragraph,  relates  to  pinions  having  the  working  face  exclusively 
of  rawhide.  Pinions  are,  however,  frequently  constructed  with 
bronze  flanges  on  the  sides  having  teeth  cut  through  the  flanges, 
these  forming  part  of  the  working  face.  In  that  case,  the  strength 
of  the  pinion  would  be  increased  from  10  to  25  per  cent,  accord- 
ing to  the  grade  of  bronze  used  and  the  thickness  of  the  flanges. 

General  Remarks  Relating  to  the  Use  of  Rawhide  Gears.  — 
An  important  consideration  in  the  use  of  rawhide  pinions  is  that 
rigid  supports  are  employed.  If  there  is  excessive  vibration, 
due  either  to  poor  supports  of  the  machines  themselves  or  to 
insufficient  bearings,  the  disalignment  of  the  meshing  teeth  that 
results  causes  excessive  wear.  In  one  extreme  case,  a  pinion 
mounted  on  an  improperly  supported  shaft  of  a  motor  of  75 
horsepower  was,  after  twenty-four  hours'  continuous  running, 
worn  away  to  about  two- thirds  of  the  original  tooth  thickness. 
After  having  provided  ample  bearings  for  the  shaft  and  installed 
a  new  pinion,  this  latter  was  in  service  for  two  and  one-half  years 
doing  good  work. 

In  a  train  of  gears  only  one  should  be  made  from  rawhide,  the 
other  being  of  metal.  Usually  rawhide  is  employed  for  the 
smaller  gear  or  the  pinion,  because  the  rawhide  is  somewhat 
more  expensive  than  metal.  The  mating  gear  should  have 
accurately  cut  teeth,  because  the  rough  surface  of  cast  teeth  will 
wear  into  the  rawhide  and  irregular  spacing  will  produce  a  bad 
strain  upon  the  rawhide  teeth.  The  New  Process  Gear  Corpora- 


STRENGTH  AND   DURABILITY  65 

tion  advises  that  pinions  with  flanges  extending  to  the  periphery 
of  the  teeth  should  be  used  wherever  the  duty  is  severe,  as  this 
construction  prevents  the  outer  layers  of  the  rawhide  from 
curling  over  and  thus  weakening  the  teeth.  Rawhide  gears  of 
fairly  large  size  are  sometimes  made  having  a  cast-iron  center  or 
spider,  with  a  rawhide  annular  ring  enclosed  on  the  sides  by  pro- 
jecting cast-iron  flanges.  In  this  way,  the  teeth  are  made  from 
rawhide  while  the  body  or  framework  of  the  gear  is  made  from 
cast  iron,  thus  producing  a  strong  combination  which  is  not  ex- 
cessively expensive  and  has  all  the  advantages  of  a  rawhide  gear. 

German  Rule  for  Rawhide  Gears.  —  The  standard  German 
engineers'  handbook,  "Hlitte,"  gives  a  rule  which  may  be  trans- 
lated into  the  following  form  for  English  measurements :  To  find 
the  allowable  load  in  pounds  at  the  pitch  line  for  a  rawhide  pinion, 
multiply  the  width  of  face  in  inches  by  from  180  to  360,  and  divide 
the  product  by  the  diametral  pitch.  It  will  be  seen  that  this  gives 
much  lower  permissible  loads  than  does  the  New  Process  Gear 
Corporation's  rule,  which  reduces  to  a  factor  of  about  470,  in 
place  of  the  180  to  360  given  in  "Hiitte."  In  both  of  these  rules 
the  strength  is  made  independent  of  the  velocity  at  the  pitch 
line,  as  already  referred  to.  Since  decrease  of  strength  with 
increase  of  velocity  is  due  to  impact,  and  since  rawhide  is  a 
substance  peculiarly  fitted  to  absorb  impact  harmlessly,  it  is 
logical  to  assume  that  the  effect  of  increasing  the  velocity  is 
negligible.  This  accounts  for  the  fact  that  a  rawhide  gear  will 
be  as  strong  as  a  cast  iron  one  at  high  speeds,  when  it  would 
appear  very  weak  in  comparison  with  it  in  a  static  test. 

Durability  of  Gearing.  —  A  pair  of  gears  figured  by  the  rules 
and  formulas  in  the  preceding  pages,  so  that  they  will  be  strong 
enough  for  the  service  for  which  they  are  to  be  used,  may  not  be 
so  proportioned  as  to  be  commercially  durable.  By  "commer- 
cially durable"  gears,  we  mean  those  which  will  last  well  in 
comparison  with  the  rest  of  the  machine  of  which  they  are  a  part. 
In  some  classes  of  machinery,  gears  strong  enough  for  their  work 
would  certainly  be  commercially  durable.  A  rack  and  pinion, 
for  instance,  used  to  raise  a  sluice  gate  for  a  dam,  if  made  strong 
enough,  would  evidently  wear  indefinitely,  though  they  might 


66  SPUR  GEARING 

rust  away.  It  is  plain  that  all  gearing  designed  for  occasional 
or  intermittent  use,  even  under  heavy  loads,  is  strong  enough  to 
wear  well  if  it  is  strong  enough  to  bear  the  load  placed  upon  it. 
With  gearing  used  for  the  continuous  transmission  of  power, 
however,  we  cannot  be  sure  of  this.  The  gearing  of  a  drive 
connecting  a  motor  with  a  printing  press,  for  instance,  might 
conceivably  be  strong  enough  and  yet  not  wear  as  long  as  the 
rest  of  the  machine. 

The  pinion  will  naturally  wear  faster  than  its  mate,  since  each 
of  its  teeth  is  in  action  a  greater  number  of  times  per  minute.  To 
make  the  life  of  the  two  more  nearly  alike,  it  is  customary  to 
make  them  of  different  materials,  as  already  mentioned,  the 
pinion  being  made  of  the  more  durable  one.  Thus,  a  combina- 
tion of  steel  pinion  and  cast-iron  gear  is  common  and  occasion- 
ally conditions  are  found  which  warrant  the  expense  of  a  hardened 
steel  pinion  and  a  phosphor-bronze  gear.  The  use  of  the  better 
material  in  the  smaller  gear  of  the  pair  is  proper  from  the  stand- 
point of  strength  as  well  as  from  that  of  durability.  An  examina- 
tion of  the  Lewis  outline  constants,  as  tabulated  in  the  preceding 
section  of  this  chapter,  will  show  that  the  teeth  of  the  pinion 
are  always  weaker  than  those  of  the  gear;  so  it  is  necessary,  if 
an  excess  of  strength  is  to  be  avoided  in  the  gear,  to  make  the 
pinion  of  the  stronger  material;  but  if  the  pinion  is  a  little  less 
durable  than  the  gear,  it  will  take  most  of  the  wear;  and  being 
more  cheaply  renewed  than  its  larger  mate,  the  mechanism  is 
kept  up  at  a  less  expense.  It  is  not  wise  to  use  soft  steel  in  both 
members  for  heavy  service  at  high  speed. 

Where  the  velocity  ratio  is  not  extreme,  but  severe  service  is 
exacted,  as  in  automobile  gearing,  the  two  members  may  be  made 
of  the  same  material  —  hardened  or,  preferably,  casehardened 
alloy  steel. 

Efficiency  of  Standard  Spur  Gears.  —  The  efficiency  of  two 
spur  gears  (or  of  any  other  power  transmitting  mechanism,  for 
that  matter)  is  measured  by  the  percentage  they  deliver  of  the 
power  entrusted  to  them.  Thus,  if  a  water  wheel  delivers  160 
horsepower  to  the  driving  pinion  on  the  shaft  on  which  it  is 
mounted,  and  the  mating  gear  on  the  jack-shaft  transmits  140 


STRENGTH  AND   DURABILITY  67 

horsepower  to  that  shaft,  the  efficiency  of  the  gearing  is  140  -f- 
160  =  87  £  per  cent. 

To  obtain  the  maximum  of  efficiency,  attention  must  be  paid 
to  the  following  considerations: 

Form  the  teeth  as  near  to  the  perfect  theoretical  shape  as  good 
workmanship  will  bring  them,  giving  the  acting  surfaces  a  fine 
smooth  finish.  If  the  teeth  are  milled  to  shape,  special  cutters 
should  be  used,  made  accurately  to  shape  for  the  exact  number  of 
teeth.  The  gears  must  be  mounted  firmly  and  accurately  in 
their  working  position. 

Use  hard,  close-grained  materials,  preferably  different  for  the 
two  gears.  Hardened  steel  on  phosphor-bronze  will  probably 
give  the  best  results,  though  it  is  difficult  to  be  sure  of  the  exact 
shape  in  hardened  gears,  unless  they  are  finished  by  grinding  after 
hardening.  Soft  steel  on  soft  steel  is  probably  the  worst  com- 
bination so  far  as  efficiency  is  concerned,  though  it  is  stronger 
than  cast  iron  on  cast  iron. 

Provide  continuous  and  copious  lubrication,  preferably  by  an 
oil  or  grease  bath  in  an  enclosed  casing.  Lubricated  gears  should 
always  be  enclosed  if  they  are  exposed  to  dust  or  grit  in  the 
slightest  degree,  otherwise  they  will  grind  each  other  away,  and 
might  rather  run  entirely  dry. 

Use  as  fine  a  pitch  as  possible,  without  requiring  too  wide  a 
face  to  transmit  the  power  required.  The  smaller  the  pitch,  the 
greater  the  efficiency.  Anything  that  tends  to  shorten  the  line 
of  contact,  confining  it  to  the  vicinity  of  the  pitch  point,  increases 
the  efficiency,  as  there  is  more  rubbing  at  the  beginning  and  end 
of  contact  than  when  the  teeth  are  passing  the  pitch  point. 

In  general,  it  may  be  said  that  there  is  no  method  of  transmit- 
ting power  between  two  parallel  shafts  that  is  more  efficient  than 
a  pair  of  well  designed  and  constructed  gears,  working  under 
proper  conditions.  The  highest  efficiency,  of  course,  is  obtain- 
able only  at  a  considerable  expense,  so  judgment  is  required  to 
know  how  far  it  is  wise  to  carry  the  possible  refinements. 

Variation  of  the  Strength  of  Gear  Teeth  with  the  Velocity.  — 
The  generally  accepted  formula  for  calculating  the  strength  of 
gear  teeth  is,  as  already  mentioned,  that  proposed  by  Mr.  Wil- 


68 


SPUR  GEARING 


fred  Lewis.  The  merit  of  this  formula  lies  in  the  great  number 
of  variables  taken  into  account  as  compared  with  other  rules  in 
more  or  less  common  use,  and  in  the  fact  that  these  variables  are 
rationally  considered.  The  effect  of  each  of  them  can  be  calcu- 
lated with  some  assurance,  with  the  single  exception  of  the  in- 
fluence of  the  velocity  on  the  safe  stress.  In  the  twenty  odd 
years  since  the  formula  was  first  proposed,  the  original  values  for 
the  stress  as  affected  by  the  velocity  have  been  largely  used. 


40,000 
35,000 

«   30,000 
Ul 

cr 
£  25,000 

cc 
at 
£0   20,000 
a. 

2 
=   15,000 

x 

|   10,000 
5,000 

0 

A  =  IMAGINARY  NON-DEFLECTING  MATERIAL  AND  PERFECT  TOOTH  SHAPE. 
B----  SHOCK  ABSORBING  MATERIAL  SUCH  AS  RAWHIDE. 
C  =  TEETH  OF  CAST  IRON  AND  PERFECT  TOOTH  SHAPE. 
D  =  TEETH  OF  CAST  IRON  AND  COMMERCIAL  ACCURACY. 
E  =  TEETH  OF  CAST  IRON  AND  POOR  WORKMANSHIP. 

/ 

/ 

y 

/ 

/ 

/ 

/ 

D^ 

^ 

S 

/ 

^^j—  *• 

^- 

^^ 

/ 

/ 

JLT^**' 

^-" 

•^ 

^ 

^ 

" 

^ 

== 

—  •= 


^'              

' 

_                    ~ 


MM^B9= 

—                     - 

•LI  • 

•         '— 

C 

—            '• 

A 

. 
-R- 

VELOCITY  AT  PITCH  LINE  IN  FEET  PER  MINUTE 
Machinery 

Fig.  4.    Hypothetical  Diagram  showing  the  Relation  of  the  Velocity  to 
the  Fiber  Stress 

Many  designers,  however,  have  felt  that  these  values  are  rather 
unsatisfactory,  although  most  of  them  will  agree  that  they  err 
rather  on  the  side  of  safety  than  otherwise.  By  referring  to  Mr. 
Lewis'  original  paper  it  will  be  seen  that  these  values  were  not 
given  as  being  definitely  determined,  but  merely  as  agreeing  well 
with  successful  cases  met  with  in  his  own  practice.  The  following 
is  a  general  analysis  of  the  conditions  involved. 

Variation  in  Strength  due  to  Impact.  —  A  variation  in  the 
strength  of  the  teeth  of  a  gear,  due  to  a  variation  in  the  velocity, 
can  be  due,  of  course,  to  but  one  thing  —  impact.  To  illustrate 


STRENGTH  AND   DURABILITY  69 

this  idea,  and  to  show  the  cause  of  the  impact,  we  will  study  the 
action  of  gearing  under  three  different  conditions.  First,  when 
made  of  an  imaginary  material  which  does  not  deflect  under  any 
strain  below  the  breaking  point.  Second,  with  gears  of  com- 
mercial material,  such  as  steel,  with  teeth  of  perfect  form.  Third, 
gears  of  commercial  material  with  teeth  of  commercial  accuracy. 

1 .  Gears  of  an  imaginary  undeflectable  material.     In  Fig.  4  is 
a  diagram  in  which  the  horizontal  distances  give  velocity  in  feet 
per  minute,  and  vertical  distances  give  stresses  in  pounds  per 
square  inch,  starting  in  this  case  at  4000,  which  is  assumed  to  be 
the  maximum  fiber  stress  in  the  gear  we  are  considering,  due  to 
the  load  at  the  pitch  line,  which  is  supposed  to  be  constant  at  all 
speeds.     If  the  teeth  of  this  gear  are  perfectly  formed  and  well 
fitted  together,  so  that  there  is  no  backlash,  if  the  power  is  de- 
livered to  them  steadily  and  smoothly,  and  the  mechanism  they 
drive  runs  without  shock,  any  disturbance  of  the  even  move- 
ment will  be  impossible,  and  impact  will  be  entirely  absent.     In 
the  diagram  in  Fig.  4,  then,  there  will  be  no  rise  of  maximum 
fiber  stresses  with  the  velocity,  so  that  the  horizontal  line  A  will 
show  the  conditions  for  this  imaginary  case. 

2.  With  commercial  material  and  theoretically  accurate  work- 
manship.    The  conditions  in  this  case  are  shown  in  Fig.  5,  with 
all  the  phenomena  greatly  exaggerated.     The  full  lines  show 
the  conditions  under  load,  while  the  dotted  outlines  show  the 
conditions  when  the  load  is  removed  from  the  driven  gear.     The 
teeth  AI,  BI  and  A2,  Bz,  carrying  the  load,  are  deflected  by  it,  as 
shown.     Tooth  B,  just  about  to  come  into  contact  with  tooth  A , 
is  on  that  account  shifted  from  its  normal  position;  it  should  be 
located  as  shown  by  the  dotted  lines.     If  it  were  in  this  position, 
it  would  come  in  contact  with  tooth  A  under  mathematically  per- 
fect conditions,  and  there  would  be  no  shock  of  engagement. 
As  it  is,  the  two  come  suddenly  into  action  as  shown  at  E,  under 
different  conditions  than  those  contemplated  by  the  design,  thus 
the  contact  takes  place  in  the  form  of  a  slight  blow,  after  which 
the  teeth  are  deflected  more  and  more,  until  they  have  taken  up 
their  share  of  the  load,  as  shown  later  at  AI  and  BI.     If  the  gears 
are  moving  very  slowly,  the  deflectior;  takes  place  very  slowly, 


70  SPUR   GEARING 

and  the  problem  is  practically  a  static  one.  If  the  gears  are 
running  at  a  high  velocity,  the  problem  becomes  essentially  a 
dynamic  one,  and  the  stresses  induced  are  greater  than  with  the 
slow  speed. 

The  increase  in  stress  with  the  increase  in  speed  for  this  second 
case  could  probably  be  represented  by  a  line  something  like  C, 
in  Fig.  4.  The  location  of  this  line  is  purely  hypothetical.  All 
we  can  say  about  it  is  that  the  increase  in  stress  as  the  speed  is 
increased  would  be  comparatively  small,  and  probably  regular. 
The  line  has  been  drawn  straight  for  convenience;  we  do  not 
know  what  the  real  shape  is. 

3.  With  commercial  materials  and  commercial  accuracy. 
This  is,  of  course,  the  practical  case  to  consider.  A  line  to  show 
the  relation  of  the  velocity  to  the  maximum  fiber  stress  for  a 
given  gear  would  very  probably  look  something  like  D  in  Fig.  4. 
This  is,  in  fact,  approximately  the  line  which  embodies  the  con- 
clusions of  the  Lewis  tables  for  a  static  stress  of  4000  pounds. 
It  is  considerably  higher  than  line  C,  because  impact  due  to 
irregular  tooth  outlines  is  added  to  the  impact  due  to  the  deflec- 
tion. In  all  probability  the  latter  is  comparatively  unimportant 
as  compared  to  that  due  to  irregularity  of  outline  in  gears  of  only 
ordinary  workmanship. 

Deflection  and  Stresses  caused  by  Impact.  —  It  may  be 
objected  that  the  deflections  produced  either  by  the  gears  coming 
into  mesh  out  of  step,  as  in  case  No.  2,  or  with  the  added  aggra- 
vation of  poor  workmanship,  as  in  case  No.  3,  are  so  minute  that 
they  could  scarcely  be  considered  as  a  serious  factor  in  the  prob- 
lem. It  is  true  that  these  deflections  are  minute  —  undetectable 
even,  by  ordinary  means;  but  this  admission  does  not  destroy 
the  argument  for  laying  to  this  distortion  the  increase  of  the 
stress  with  the  speed.  If  great  loads  produce  slight  deflections, 
slight  deflections  likewise  produce  great  stresses,  so  that  the 
slight  bending  brought  about  by  the  teeth  coming  into  contact 
at  E  in  Fig.  5,  under  slightly  imperfect  conditions,  may  produce 
great  effects  proportionately  in  the  fiber  stress,  and  the  effects 
are  magnified  by  the  irregularities  due  to  poor  workmanship. 
When  we  stop  to  figure  out  what  load  per  inch  of  face  is  re- 


STRENGTH  AND  DURABILITY 


quired  to  deflect  a  2-inch  circular  pitch  gear,  say  0.003  inch,  it  is 
evident  that  an  irregularity  in  outline  of  this  amount  would 
scarcely  be  negligible  at  high  speeds,  if  our  hypothesis  is  correct. 
The  phenomena  of  impact  are  complicated  to  a  high  degree. 
The  maximum  stresses  produced  depend  on  the  rapidity  of 
transmission  of  a  wave  of  stress  or  deflection  produced  in  the 
material  by  the  impact.  If  this  wave  is  propagated  slowly,  the 
stresses  are  high;  if  rapidly,  the  stresses  are  low.  The  factors 
entering  into  the  problem  are  the  elasticity  of  the  material,  and 
the  mass  and  shape  of  the  part  affected.  In  very  simple  cases 


FULL  LINES  SHOW  CONDITIONS 

UNDER  LOAD. 

DOTTED  LINES  SHOW  CONDITIONS 
UNDER  NO  LOAD. 


Machinery 


Fig.  5.    Action  of  Gear  Teeth  under  Load,  greatly  Exaggerated 

the  problem  has  been  investigated  mathematically,  but  our 
problem  with  the  gear  teeth  is  so  complicated  that  we  must  of 
necessity  at  once  apply  to  the  engineer's  court  of  last  resort — 
experiment. 

Practical  Considerations  Affecting  the  Effect  of  Shocks.  —  It 
is  evident  that  other  variables  besides  the  strength  of  the  material 
and  the  velocity  at  the  pitch  line  enter  into  the  fixing  of  the  line 
on  the  diagram  of  Fig.  4.  In  addition,  the  following  points  will 
have  to  be  considered. 

i.  Accuracy  of  tooth  outlines.  From  what  has  just  been  said, 
it  is  evident  that  the  variation  of  the  stress  with  the  velocity 


72  SPUR   GEARING 

will  be  affected  by  the  accuracy  of  the  workmanship  involved 
in  forming  the  tooth  of  the  gear.  Investigating  the  conditions 
in  the  case  of  a  second  pair  of  gears,  similar  to  those  from  which 
line  D  was  determined,  but  of  a  considerably  poorer  grade  of 
workmanship,  we  should  expect  to  find  results  giving  a  line 
something  like  E  on  the  same  diagram,  giving  much  higher  values 
for  the  stresses  resulting  from  the  load.  It  is  evident,  then,  in 
considering  lines  C,  D  and  E  that  workmanship  is  a  variable 
which  should  be  considered  in  the  experiments,  and  that  a  series 
of  tests  should  be  run  with  two  sets  of  gears  of  varying  workman- 
ship, one  of  high  and  the  other  of  only  ordinary  grade,  to  make 
sure  that  this  consideration  is  really  of  importance. 

2.  Design  of  gear  and  mechanism.      Another  factor  which 
may  affect  the  increase  of  the  stress  with  the  speed  is  the  design 
of  the  rim  and  spokes  of  the  wheel.     It  is  conceivable  that  a  gear 
with  a  very  heavy  rim  and  rigid  spokes  will  absorb  the  shocks  due 
to  high  velocity  less  easily  than  a  gear  with  a  light  rim  and  flexible 
spokes  or  arms.     The  whole  structure  of  the  machine  in  which 
the  gearing  is  carried,  so  far  as  its  rigidity  and  massiveness  are 
concerned,  should,  in  fact,  affect  this  matter.    The  further  away 
from  the  point  of  tooth  contact  the  members  of  the  structure 
are,  however,  the  less  effect  will  they  have,  so  perhaps  even  the 
influence  of  the  arms  and  rim  can  be  neglected.     The  same  con- 
sideration affects  the  design  of  the  mechanism  to  be  used  in  the 
tests.     It  is  conceivable  that  a  mechanism  involving  long  shafts 
and  other  flexible  members  might  give,  for  a  given  set  of  gears,  a 
line  lower  down  on  the  diagram  of  Fig.  4  than  would  be  the  case 
if  the  construction  were  very  heavy  and  rigid.     The  supporting 
mechanism  must  not  in  any  case,  of  course,  deflect  in  such  a  way 
as  to  prevent  the  teeth  from  having  a  full  bearing  on  each  other. 

3.  The  nature  of  the  materials  used.    Referring  to  what  has 
previously  been  said  as  to  the  factors  governing  impact,  it  will 
be  seen  that  the  nature  of  the  material  used  would  affect  the 
shape  of  the  curve.     It  is  probable,  for  instance,  that  two  sets 
of  gears,  one  made  of  cast  iron  and  the  other  of  a  bronze  alloy 
of  the  same  tensile  strength,  would  show  lines  of  very  different 
shape,  owing  to  the  difference  in  the  modulus  of  elasticity  and 


STRENGTH  AND   DURABILITY  73 

the  specific  weight  of  these  two  substances.  From  this  it  will 
be  seen  that  we  cannot  be  sure  that  the  results  found  to  be 
applicable  to  cast  iron  or  steel  could  also  be  applicable  to  either 
a  pair  of  bronze  gears  or  to  the  case  of  a  bronze  gear  meshing 
with  a  mate  of  steel  or  iron.  That  the  nature  of  the  material 
would  have  a  vital  effect  on  the  shape  of  the  curve  is  still  more 
probable,  when  we  consider  the  practice  followed  in  the  use  of 
such  substances  as  rawhide.  This  material  is  particularly  fitted 
to  sustain  impact  and  absorb  without  undue  stress  the  deflection 
caused  by  it.  Owing  to  this  characteristic,  we  might  expect  that 
the  line  for  a  gear  of  this  substance  would  be  practically  hori- 
zontal, as  shown  at  B  in  the  diagram,  approaching  A,  though 
governed  by  entirely  different  conditions  from  those  producing 
A.  So  far  as  can  be  learned,  this  supposition  agrees  with  the 
practice  of  the  manufacturers  of  rawhide  gears. 

It  may  be  found  that  the  points  mentioned  have  so  much 
influence  on  the  question  at  issue  that  it  would  be  very  difficult  to 
lay  down  a  law  governing  the  variation  of  stress  with  velocity, 
and  that  the  most  that  can  be  done  is  to  determine  the  varia- 
tion in  cases  of  commercial  workmanship  and  rigid  design, 
using  the  relation  thus  established  in  an  empirical  formula, 
with  the  knowledge  that  poorer  conditions  may  bring  the  fiber 
stresses  much  higher,  while  good  workmanship  and  careful 
design  may,  on  the  other  hand,  bring  them  much  lower.  Quite 
possibly  the  factors  now  in  use  may  be  found  to  nearly  fill 
commercial  requirements,  in  which  case  we  must  conclude  that 
the  criticisms  of  their  being  too  high  have  been  founded  on 
experience  with  cases. combining  the  favorable  conditions  just 
mentioned. 

Practical  Considerations  Affecting  Design.  —  The  fact  that 
the  variation  of  the  strength  with  the  velocity  is  due  to  impact 
suggests  also  a  number  of  points  relating  to  design.  Most  of 
these  are  already  well  known,  and  are  standard  practice,  the 
conclusions  being  so  obvious  that  simple  common  sense  has 
suggested  them  without  theoretical  analysis  being  necessary. 

i.  Value  of  accuracy.  It  is  evident  that  this  theory  of 
impact  puts  a  premium  on  accuracy  in  workmanship  for  gears 


74  SPUR   GEARING 

that  are  to  run  at  high  speed  under  a  heavy  load.  It  is  probable 
that  the  strength  of  a  given  pair  of  gears  may  be  cut  in  two  if  the 
tooth  outlines  are  not  carefully  determined,  and  if  the  cutter  is 
not  set  centrally.  This  suggests  the  desirability  of  a  greater 
sub-division  of  the  standard  cutter  series  for  work  of  this  kind. 
Of  course,  the  gears  can  always  be  made  heavy  enough  for  the 
required  service,  but  the  extra  cost  of  accurate  cutters  and 
careful  cutting  will  be  repaid  in  cases  where  light  weight  and 
compact  design  are  at  a  premium.  In  such  cases  the  use  of 
cutters  specially  designed  for  each  gear  is  recommended. 

2.  Resilience  of  design  and  materials.     In  high-speed  gearing 
it  is  evident  that  the  shock  due  to  the  impact  should  be  absorbed 
as  quickly  and  as  fully  as  possible.     This  suggests  the  use  at 
abnormally  high  speeds  of  rawhide,  cloth,  etc.,  for  one  of  the 
members  of   the  pair  of  gears.     The  introduction  of  spring 
couplings  or  similar  devices  may  also  be  desirable,  especially 
where  the  other  parts  of  the  mechanism  are  liable  to  transmit 
shock  to  the  gearing. 

3.  Easing  of  the  points  of  the  tooth.     There  has  always  been 
a  sort  of  superstition  that  the  points  of  the  tooth  should  be  eased 
off  to  make  the  action  smoother.    This  is  done,  of  course,  in 
standard  involute  gears,  though  for  another  reason,  that  of 
avoiding  interference  with  the  flanks  of  the  pinions.     It  can 
now  be  seen  that  there  is  a  solid  basis  for  this  practice  in  all  cases 
where  gears  are  to  run  at  such  speeds  that  severe  impact  is 
liable  to  take  place.     Referring  to  Fig.  5,  teeth  A  and  B  are 
taking  up  the  load  very  suddenly,  owing  to  the  fact  that  they 
are  out  of  step,  due  to  the  deflection  of  the  other  teeth  momen- 
tarily carrying  the  load.     Easing  away  the  points  of  A  and  B 
would  mitigate  this  sudden  reception  of  the  load,  allowing  the 
inevitable  deflection  to  take  place  more  slowly,  with  a  consequent 
gain  in  the  strength  of  the  gear  at  high  speeds.     It  would  have 
a  similar  effect  in  minimizing  impact  due  to  inaccuracy  of  out- 
line.    This  modification  of  the  outline  of  the  tooth  should  be 
very  slight,  and  extend  but  a  short  distance,  so  that,  when  the 
load  is  entirely  transferred,  the  "eased  off"  portion  of  the  curve 
will  be  passed,  and  the  true  involute  or  cycloidal  portion  begun. 


CHAPTER  IV 
SIMPLIFIED  FORMULAS  FOR  STRENGTH  OF  GEARS 

Need  for  Simplified  Formulas.  —  It  is  generally  conceded  that 
the  Lewis  formula  for  the  strength  of  gear  teeth,  with  its  accom- 
panying tables,  is  the  most  accurate  in  form,  as  the  maximum 
strength  of  each  tooth  is  determined  from  its  shape.  It  may  be 
safely  used  for  determining  the  strength  of  gears  made  by  modern 
methods,  but  its  tabulated  form  makes  it  difficult  to  use  from 
the  standpoint  of  the  designer.  It  is  well  adapted  to  determine 
the  strength  of  any  given  gear  or  pinion.  But  the  reverse 
process  —  that  of  finding  a  gear  suitable  to  meet  the  condition 
of  a  given  horsepower  and  revolutions  per  minute  —  is  not  so 
simple,  the  trial-and-error  method  being  a  lengthy  one  at  best. 
The  following  deductions  give  close  and  rapid  approximations  for 
preliminary  work. 

Assumptions  on  which  Formulas  are  Based.  —  As  both  the 
gear  and  its  pinion  are  usually  made  of  the  same  material,  either 
cast  iron  or  cast  steel,  the  strength  of  the  pair  is  determined  by 
the  strength  of  its  weakest  member,  which  is  the  pinion  when 
made  of  the  same  metal  as  the  gear.  For  economical  reasons 
the  pinion  is  usually  limited  to  about  15  teeth,  so  we  may  take 
that  number  as  a  convenient  base.  Circular  pitch  is  used  in  the 
calculations,  but  the  circular  pitch  can  finally  be  transformed 
into  diametral  pitch  if  this  is  desired. 

In  a  train  of  gears,  the  maximum  reduction  on  any  pair  is 
usually  taken  at  4  or  5  to  i,  so  the  number  of  reductions  and 
ratios  may  be  quickly  deduced.  Then  the  problem  is  usually 
presented  as  follows: 

Given  the  horsepower  and  revolutions  per  minute  of  the  pinion, 
what  will  be  the  allowable  working  stress,  pitch,  face,  factor  of 
strength  and  diameter? 

The  majority  of  trade  gear  lists  give  the  horsepower  of  gears  at 

75 


76  SPUR   GEARING 

100  R.P.M.  with  an  allowable  stress  for  cast  iron  of  3000  pounds 
per  square  inch;  but  it  is  more  difficult  to  transform  this  horse- 
power to  suit  the  other  conditions,  than  to  proceed  independently. 

Values  of  Safe  Working  Stress.  —  The  values  of  S,  the  safe 
working  stress,  which  Mr.  Lewis  adopted  tentatively,  as  they 
gave  satisfactory  results  in  practice,  were  as  follows.  (See  also 
table  in  preceding  chapter,  where  these  values  are  slightly 
modified.)  • 

Let  V  =  speed  of  teeth  in  feet  per  minute  and  5  =  safe  work- 
ing stress,  then: 

For  V  —  100  (or  less)  200  300  600  900  1200  1800  2400 
For  cast  iron: 

S  =  8000  6000  4800  4000  3000  2400  2000  1700 
For  cast  steel: 

S  =  20,000  15,000  12,000  10,000  7500  6000  5000  4300 

When  these  values  are  plotted,  it  will  be  seen  that  the  curves, 
though  slightly  irregular,  closely  approximate  curves  of  the 
hyperbolic  form.  The  equations  of  the  curves  which  most  nearly 
agree  with  the  Lewis  values  are  found  to  be  the  following: 

88,000 
For  cast  iron,  S  = — '-=- . 

Vv 

-^  ,    0      220,000 

For  cast  steel,  S  =  • — *=r- . 

VV 

These  formulas  give  the  following  comparative  values: 
When  V  =      100       200       300    600    900  1200  1800  2400 
For  cast  iron: 

S  =   8800     6250      5000  3600  2930  2540  2080  1790 
For  cast  steel: 

S  =     22,000  15,625  12,500  9000  7325  6350  5200  4475 

The  agreement  with  the  Lewis  assumed  values  is  remarkably 
close.  The  new  values  will  probably  come  much  nearer  the  true 
ones,  as  they  are  in  much  better  line.  They  are  also  much  more 
dependable,  as  the  stress  suitable  for  any  speed  can  be  easily 
found  from  the  formula  to  the  fraction  of  a  pound,  if  desired,  on 


FORMULAS  FOR   STRENGTH  77 

a  true  curve  ;  whereas,  the  use  of  the  tabular  values  results  in  the 
substitution  of  values  which  descend  by  variable  steps  of  from 
2000  to  300  pounds  at  a  jump,  or  if  ordinary  interpolation  is  used 
the  result  is  still  inaccurate,  as  the  interpolation  necessarily 
follows  a  straight  line  between  the  two  nearest  values,  and  is  thus 
too  high.  The  new  curve  values  also  come  nearer  to  the  com- 
parative Harkness  values  as  given  by  Kent. 

Derivation  of  Simplified  Formulas.  —  The  face  of  gears,  A,  is 
another  variable  quantity;   but  in  the  manufacturer's  standard 
lists  of  today  the  face  is  usually  about  three  times  the  pitch,  and 
this  may  be  adopted  as  close  enough  for  preliminary  work.     It 
will  be  found  that  the  majority  of  stock  gears  have  either  14^- 
degree  involute  or  cycloidal  teeth,  so  these  styles  will  be  used  in 
these  calculations.     The  factor  of  strength,   Y',  in  the  Lewis 
tables  for  a  i5~tooth  pinion  of  these  types  is  0.075.     The  factor 
y'  is  found  from  the  values  of  factor  Y  in  the  table  "Factors  for 
Calculating  Strength  of  Gear  Teeth/'  in  the  preceding  chapter, 
by  dividing  the  value  of  Y,  as  given,  by  3.14.     We  have,  there- 
fore, the  following  data  for  a  i5-tooth  cast-iron  spur  pinion: 
Let       S  =  safe  working  stress,  in  pounds; 
P'  =  circular  pitch,  in  inches; 
A  =  face,  in  inches; 
Y'  =  factor  of  strength; 
V  =  speed  of  pitch  line,  in  feet  per  minute. 
The  Lewis  general  formula  reduces  to 
Hp   _SP'AY'V 

33,000 
From  our  average  determination  above,  we  have: 

0  88.OOO  TV  Tr/ 

S  =  -=r  ;    A=$P']    Y'  =  0.075. 


Substituting  these  values  in  the  general  formula  and  reducing, 
we  have  for  a  i5-tooth  cast-iron  spur  pinion: 

H.P.  =o.6P'2  W     .    ..-.'.  ...     (i) 
By  a  similar  process,  we  find  for  a  i5-tooth  cast-steel  spur 
pinion: 

H.P.  =  1.5  P'2  Vv    ......     (2) 


78  SPUR    GEARING 

For  a  bevel  pinion,  let 

d  =  small  diameter  of  bevel; 
D  =  large  diameter  of  bevel. 

cp'  A  W      d 

Then  H.P.  =*F  Ay  y-  X- 

33,000         D 

As   r  usually  equals  about  -  ,  we  can  say: 


3 
and  for  a  i5-tooth  cast-iron  bevel  pinion, 

H.P.  =  0.4  P'2  W.     ......     (3) 

For  a  i5-tooth  cast-steel  bevel  pinion, 

H.P.  =P'*VV.    ...    .    ._."  .     (4) 

We  now  wish  to  find  V  in  terms  of  revolutions  per  minute. 
For  a  i5~tooth  pinion,  approximately: 

V  =  15  X  R.P.M.  X  J"  .     f   R  p  M       p, 

12 

Substituting  this  value  in  Formula  (i)  we  have: 


H.P.  =  0.6  P/2  Vi.25  R.P.M.  X  P'. 
Squaring,  H.P.2  =  0.36  P/4  (1.25  R.P.M.  X  P')- 

Reducing,  and  solving  for  P' ',  we  have  for  a  cast-iron  spur 
pinion : 

p/=    V^ILR* 
V    R.P.M 

A  similar  substitution  and  reduction  in  Formulas  (2),  (3)  and 
(4)  gives  the  following:  

For  cast-steel  spur  pinion,  Pf  =  V  °  p  p  M     •    •     •  (6) 

For  cast-iron  bevel  pinion,  P'  =  V   j?  p  M    •     •     •     (7) 

Vo  8  H  P  2 
For  cast-steel  bevel  pinion,  P'  =  y    '   p  M    •    •     •     (8) 


FORMULAS   FOR  STRENGTH 


79 


For  rapidly  varying  loads,  or  where  there  is  much  starting  and 
stopping,  it  is  well  to  reduce  the  safe  stress  to  two-thirds  that 
allowed  by  the  above  formulas.  We  then  have: 

For  cast-iron  spur  pinion,  H.P.  =  o.4P'2  VF;  P'  =  VR^'M*^ 


For  cast-steel  spur  pinion,  H.P.  =  P'2  VF;  P' =  V  "jf^f  (I0) 

5/TT   OHP2 

For  cast-iron  bevel  pinon,  H.P.  =0.27  P'2  VV;  P' =  V   RPM' 


For  cast-steel  bevel  pinion,  H.P.  =  o.6?P'2VV'9  P'  = 

R..P.JM. 

(12) 

The  fifth  root  can  be  easily  determined  by  logarithms  on  the 
slide  rule,  or  from  the  usual  tables,  but  the  values  for  the  com- 
mon cases  are  given  later. 

Corrections  for  Tooth  Numbers.  —  It  now  remains  to  deter- 
mine the  correction  for  different  numbers  of  teeth.  As  the  teeth 
of  pinions  generally  range  from  12  to  30,  we  need  not  go  outside 
these  limits.  Let  N  =  number  of  teeth.  Plotting  the  Lewis 
values  for  Y'  for  this  case,  and  determining  the  nearest  curve,  we 
find  that  the  straight  line  formula: 

y,       =      2     N     +     45 

IOOO 

expresses  this  curve  very  closely,  as  will  be  seen  by  the  following 
comparative  table: 


No.  of 
Teeth,  N 

F'by 

Formula 

Y'  from 
Lewis'  Tables 

No.  of 
Teeth,  N 

F'by 
Formula 

Y'  from 
Lewis'  Tables 

12 

0.069 

0.067 

19 

0.083 

0.087 

13 

0.071 

0.070 

20 

0.085 

0.090 

14 

0.073 

0.072 

21 

0.087 

0.092 

IS 

0.075 

0.075 

23 

0.091 

0.094 

16 

0.077 

0.077 

25 

0.095 

0.097 

17 

0.079 

0.080 

2? 

0.099 

0.100 

18 

O.oSl 

0.083 

30          : 

0.105 

O.  IO2 

8o  SPUR   GEARING 

Therefore,  for  other  teeth,  we  can  multiply  the  horsepower 

2  N  +  4.^ 

given  in  the  above  formulas  by  -          -*%  or  more  briefly  by 

0.027  N  +  0.6. 

Correction  for  Increased  yelocity.  —  We  must  also  correct 
for  the  increased  velocity  of  this  larger  pinion,  i.e.,  multiply  the 

result  by  y —   or  0.26  vW.    The  continued  product  of  these 

last  two  multipliers  might  be  used,  but  this  does  not  simplify 
the  calculation.  These  corrections  need  seldom  be  applied  for 
preliminary  work. 

To  Find  the  Pinion  Diameter.  —  Lastly,  to  find  the  diameter 
of  the  pinion,  approximately: 

,.  NXP' 

diameter  =  -       — , 

7T 

or  diameter  =  0.31%  NP', 

or  for  a  15 -tooth  pinion, 

diameter  =  4.77  Pr (13) 

If  diametral  pitch  is  desired,  it  is  sufficiently  close  to  say: 

diametral  pitch  =  ^ (14) 

Summary  of  Formulas.  —  The  following  formulas,  therefore, 
Nos.  (5)  to  (14)  (as  deduced  above),  give  closely  enough  for  all 
preliminary  determinations,  the  size  of  pinion  required  for  15 
teeth.  P'  =  circular  pitch. 

Stress  from  Stress  % 

Lewis'  Tables  Lewis'  Tables 


D,         //2.22H.P.2  '/5.0H.P.2 

Cast-iron  spur  pinion,  P  =  \f     .  y 


R.P.M.  *    R.P.M. 


D,      47o.36H.P.2        //0.8H.P. 
Cast-steel  spur  pinion,  P>  =  V   pPM  V  ^VW 


....       „,      //s.oH.P.2  //II.OH.P. 

Cast-iron  bevel  pinion,  P    -  y  ^R  p  M  V    R.P.M. 

//0.8H.P.2  //I.8H.P.2 

Cast-steel  bevel  pinion,  P'  ^V^pl^  V  ^pjST 

Pitch  diameter  =  4.77  P' 
Diametral  pitch  =  ^7 


FORMULAS   FOR  STRENGTH 


8l 


Practically,  stock  gears  are  made  up  to  3  inches  circular  pitch 
by  J-inch  steps,  and  a  pitch  of  less  than  i  inch  is  seldom  used. 

The  following  table  will  therefore  determine  the  roots  for  the 
nearest  common  pitch: 


No.  or 

Fifth 

No.  or 

Fifth 

No.  or 

Fifth 

Root 

Power 

Root 

Power 

Root 

Power 

3/4 

0.24 

2 

32 

$H 

525 

I 

I 

2H 

58 

4 

1024 

IM 

3 

2H 

98 

4M- 

1845 

rW 

8 

2% 

158 

5 

3I2S 

i% 

16 

3 

243 

6 

7776 

In  case  the  revolutions  per  minute  of  the  pinion  are  less  than 
80,  which  is  exceptionally  slow,  care  must  be  taken  in  applying 
the  formula,  or  the  allowable  stress  may  be  exceeded.  With  a 
i5-tooth  pinion: 

80  R.P.M.  =  100  feet  per  minute  for  i-inch  P' . 

40  R.P.M.  =  100  feet  per  minute  for  2-inch  Pf . 

27  R.P.M.  =  100  feet  per  minute  for  3-inch  P1 '. 

20  R.P.M.  =  100  feet  per  minute  for  4-inch  P1 '. 

Chart  for  Rapid  Solution  of  Gear  Problems.  —  A  simple  three 
quadrant*  chart  has  been  prepared  for  the  rapid  solution  of  these 
problems  by  mere  inspection,  good  for  any  number  of  teeth,  and 
for  all  the  different  styles,  materials,  and  stresses  of  gears  given 
by  the  above  formulas,  but  for  occasional  preliminary  determi- 
nation, the  formulas  are  sufficient,  as  their  solution  is  simple. 

It  will,  of  course,  be  understood  that  the  teeth  considered  in 
these  formulas  are  those  of  the  usual  standard  dimensions  for 
cast  gears,  in  which  the  height  of  tooth  equals  seven-tenths  of 
the  pitch.  What  are  known  as  "  short- tooth  gears,'7  in  which  the 
height  of  tooth  equals  half  the  circular  pitch,  are  undoubtedly 
stronger,  but  their  smaller  working  face  has  by  many  been  sup- 
posed to  cause  more  rapid  wear,  and  their  use  is  not  so  common. 
Although  machine-molded  cast  gears  run  quietly  at  low  speeds, 
they  should  not  be  used  for  rim  speeds  much  over  1000  feet  per 
minute.  For  speeds  of  from  1000  to  3000  feet  per  minute  cut 
gears  should  be  substituted. 


82 


SPUR   GEARING 


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o  mula  value,  or  |  of  same);  thence  to  R.  P.  M.;  and  thence  to  pitch. 
-1  Or  reverse,  going  from  pitch  in  third  quadrant  to  H.  P.  in  first.  With 
o  15  teeth,  first  quadrant  may  be  omitted.  Example:  Find  pitch  of  cast- 
^  iron  spur  pinion  of  30  teeth,  to  transmit  40  H.  P.  at  150  R.  P.  M. 
00  Starting  in  first  quadrant,  follow  40  H.  P.  line  to  the  left  until  it  inter- 
§  sects  diagonal  for  30  teeth;  thence  vertically  to  intersection  with  curve 
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WEIGHT  AND  PRICE  83 

For  a  quick  approximation  of  the  diameter  of  the  pinion  shaft 
in  inches,  the  following  formula  may  be  used : 

shaft  diameter  =  Pf  +  i. 

Weight  of  Gears.  —  The  weight  of  pinions  and  gears  varies 
with  different  makers.  Pinions  of  from  12  to  30  teeth  are  usually 
made  slightly  wider  than  gears,  even  if  they  are  not  shrouded; 
and  the  smaller  sizes  have  solid  webs  in  place  of  arms.  It  is 
found  that  a  formula  of  the  form: 

weight  in  pounds  =  coefficient  X  P'2AN} 

will  usually  fit  the  weights. 

For  many  tables,  the  coefficients  of  the  following  values  will 
serve: 

weight  of  pinion  =  0.35  P'2AN, 
weight  of  gear  =  0.45  P'2AN, 

or  where  A  =  3  P't 

weight  of  pinion  =  P'W, 
weight  of  gear  =  1.35  P'W, 

or  when  diameter  and  P'  are  known,  as  N  =  — , 

weight  of  pinion  =  3.1  DP'2, 
weight  of  gear  =  4.2  DP'2. 

Price  of  Gears.  —  The  price  of  gears  varies  largely  with  dif- 
ferent manufacturers.  The  price  of  cast-tooth  spur  gears  can 
usually  be  expressed  by  a  formula  of  the  following  form : 

price  =  (coeff.  X  P'N)  +  (coeff.  X  P'}. 

Cut  tooth  gears  usually  cost  about  20  per  cent  more  than  cast 
tooth  gears;  and  cast-steel  gears  from  50  to  75  per  cent  more 
than  cast-iron  gears  of  the  same  size. 


CHAPTER  V 
THE  STUB-TOOTH  GEAR 

DURING  the  last  few  years,  a  form  of  gear  tooth,  known  as  the 
"stub  gear-tooth,"  has  been  introduced.  It  has  been  applied 
successfully,  especially  to  automobile  drives.  The  features  of 
this  form  of  gear  tooth  are  a  shorter  addendum  and  dedendum 
than  used  for  ordinary  standard  gears.  There  are  several 
systems  of  these  teeth  in  use,  but  the  stub  gear-tooth  introduced 
by  the  Fellows  Gear  Shaper  Company  of  Springfield,  Vt,  is  by 
far  the  most  commonly  used.  The  information  relating  to  this 
class  of  tooth,  given  in  the  following,  has  been  furnished  mainly 
by  this  company. 

Standards  for  Gear  Teeth.  —  With  the  constantly  increasing 
use  of  gears  for  transmitting  power,  the  question  of  the  correct 
shape  and  size  of  gear  teeth  becomes  of  far  greater  importance 
than  ever  before.  It  is  not  sufficient  merely  that  a  gear  be  well 
cut  and  the  teeth  properly  spaced;  the  shape  and  proportions 
of  the  tooth  itself  must  be  carefully  considered.  The  two  most 
important  features  to  be  secured  are  the  nearest  approach  to  a 
rolling  action  that  it  is  possible  to  obtain,  and  the  strongest  tooth 
that  will  meet  this  condition.  The  first  feature  includes  easy 
running  and  reduces  the  friction  to  the  lowest  point,  thus  pro- 
ducing the  least  wear  in  action. 

We  are  apt  to  think  of  the  present  standard  gear  tooth  as  one 
of  the  fixed  laws  of  mechanics,  and  as  representing  the  highest 
development  possible  in  this  line,  just  as  we  consider  the  question 
of  threads  as  fixed  by  the  Sellers  or  United  States  Standard 
forms.  But  this  is  not  the  case;  for  although  there  is  always  a 
perfectly  justifiable  reluctance  about  departing  from  a  recog- 
nized standard,  it  is  at  least  good  policy  to  know  both  sides  of 
the  question,  and  to  see  if  the  advantages  claimed  for  a  newer 
form  of  tooth  can  be  backed  up  with  positive  proof. 


STUB-TOOTH   GEARS  85 

The  so-called  standard  tooth  of  today  is  the  involute  curve 
with  the  i4|-degree  pressure  angle  or  angle  of  obliquity;  but  its 
use  is  not  as  universal  as  might  be  supposed,  and  the  newer  form 
known  as  the  " stub-tooth"  with  its  20-degree  pressure  angle  has 
become  established  to  a  degree  that  many  do  not  realize. 

The  Epicycloidal  Tooth  Form.  —  The  first  gear  teeth  worthy 
of  the  name  were  of  epicycloidal  form  and  measured  by  circular 
pitch;  and,  as  the  state  of  the  mechanic  arts  had  not  advanced 
to  the  point  of  cutting  teeth  in  a  machine,  the  gears  were  made 
with  cast  teeth.  With  the  epicycloidal  system  it  was  necessary, 
in  order  to  avoid  the  excessive  under-cutting  of  the  flank  of  the 
tooth,  to  adopt  certain  proportions  of  form  and  length  of  tooth, 
which  gave  continuous  action  when  the  gear  of  12  teeth  was 
used.  Proportions  for  the  length  of  tooth  and  other  dimensions, 
that  were  originated  at  this  time,  when  the  science  of  applied 
mechanics  was  in  its  infancy,  have  been  handed  down  to  the 
present  time  with  little  change,  regardless  of  the  fact  that  with 
the  dropping  of  the  epicycloidal  form  of  tooth  the  conditions 
have  been  entirely  changed. 

The  Involute  Tooth  Form.  —  The  adoption  of  the  involute 
form  of  tooth,  together  with  the  diametral  pitch  system  of 
measurement,  has  been  gradual  but  constant,  and  it  has  now 
practically  superseded  all  other  forms.  With  the  epicycloidal 
system  of  gear  teeth  the  mean  of  the  angle  of  obliquity  was  about 
15  degrees,  and  in  determining  the  proportions  of  the  inter- 
changeable involute  gearing,  this  angle  of  rack  was  adopted  as 
the  basis  of  the  system.  One  point  that  had  much  weight  in 
deciding  upon  the  angle  was  the  fact  that  the  sine  of  the  angle 
of  14^  degrees  was  almost  exactly  0.25,  a  proportion  which  was 
easy  for  the  millwright  to  lay  out;  and  so  the  angle  of  14 J  was 
decided  upon.  So  strongly  has  this  angle  become  impressed  on 
the  mechanic  that  it  has  become  almost  a  revered  tradition  that  a 
greater  angle  than  this  cannot  be  used  with  any  degree  of  success, 
because  of  excess  friction  and  the  consequent  wear  on  bearings. 

While  this  idea  may  have  been  more  or  less  correct  with  the 
proportions  of  shafts  and  bearings  in  use  in  former  days,  with 
cast  gears  and  epicycloidal  teeth,  it  surely  has  very  little  weight 


86 


SPUR   GEARING 


under  present  conditions.  The  effect  of  the  angle  of  obliquity 
on  the  wear  of  bearings  has  been  unduly  exaggerated;  and,  as  a 
consequence,  the  proportions  of  gearing  in  general  use  today  are 
open  to  great  improvement  in  the  three  essential  points  of 
strength,  durability  and  running  qualities- 
Length  of  Tooth.  —  With  an  interchangeable  set  of  gears  the 
length  of  the  tooth  should  be  sufficient  to  give  an  arc  of  action, 
even  with  the  smallest  pinions,  so  that  one  pair  of  teeth  will  be 
in  contact  until  the  next  pair  is  in  position  to  take  up  the  load. 
There  is,  therefore,  a  fixed  relation  between  the  length  of  tooth 


20  TEETH 

28  TEETH 


Machinery 


Fig.  i.    Automobile  Transmission  Gearing  used  as  Example  in  the 
Study  of  Stub-tooth  Gears 

and  the  angle  of  obliquity.  In  ordinary  practice  the  number 
of  teeth  of  the  pinion  is  limited  to  twelve,  and  the  length  of  the 
tooth  ordinarily  adopted  is  such  that  the  action  is  continuous. 

It  is,  however,  a  fallacy  to  argue  that  the  teeth  should  be  as 
long  as  possible,  with  the  idea  that  a  gain  is  made  if  two  or  more 
pairs  of  teeth  are  in  mesh  at  once.  Conditions  are  never  such 
that  an  equal  division  of  the  load  is  possible,  and  a  length  of 
tooth  beyond  that  which  is  necessary  to  insure  a  continuity  of 
action  produces  undue  friction  and  wear. 

Angle  of  Obliquity,  Length  of  Tooth,  Efficiency  and  Wearing 
Qualities.  —  Although  the  subject  of  gearing  has  been  investi- 


STUB-TOOTH   GEARS 


gated  and  discussed  at  great  length  by  many  mechanics,  there 
is  one  phase  of  the  subject  that  has  received  but  scant  attention, 
and  this  is  the  correlative  effect  of  the  angle  of  obliquity  (or 
pressure  angle)  and  the  length  of  the  tooth,  upon  the  efficiency 
and  wearing  qualities  of  the  tooth  itself.  It  can  be  shown  that 
an  excessive  sliding  action  takes  place  at  a  certain  portion  of  the 
tooth  action  between  gears  having  teeth  of  the  standard  or 
i4^-degree  angle,  and  that  by  increasing  this  angle  it  is  possible 
to  so  shorten  the  tooth  that  only  such  portions  of  the  curve  are 


Machinery 


Fig.  2.    Tooth  Action  of  Standard  Involute  Tooth  Gears 

used  as  will  give  nearly  a  complete  rolling  action.  An  increase 
in  the  angle  of  obliquity  has  often  been  advocated  by  others  and 
is  advantageous,  but  its  advantages  are  very  limited  if  a  tooth  of 
the  standard  length  is  retained. 

The  use  of  gears  for  the  transmission  of  power  in  automobiles 
has  perhaps  called  attention  to  this  question  more  than  any 
other  line  of  work.  To  show  clearly  just  what  this  action  really 
is,  the  accompanying  illustrations  have  been  prepared  for  com- 
parison of  the  action  of  the  stub- tooth  with  that  of  the  14^- 
degree  angle  standard  type,  using  as  an  example  the  gears  of  the 
sliding  transmission  of  an  automobile. 


88 


SPUR  GEARING 


Investigation  of  an  Actual  Design.  —  The  following  gears  have 
been  selected  as  being  typical  of  such  a  transmission  or  gear-box. 

Low  gear    18-33 

2nd  gear      26-25 

3rd  gear      20-31 

Reverse       14-20-31 

These  gears  are  shown  assembled  in  Fig.  i.  As  the  combina- 
tions 18-33  and  20-31  are  very  similar,  the  second  has  been 
omitted  from  the  comparisons  and  the  others  will  now  be  con- 
sidered. 


Machinery 


Fig.  3.    Tooth  Action  of  Stub-tooth  Gears 

In  Figs.  2  and  3  are  shown  comparisons  of  tooth  action  for 
gears  of  both  the  standard  and  of  the  stub- tooth  forms,  the  driver 
having  25  and  the  driven  26  teeth.  If,  in  the  diagrams,  the 
gears  are  supposed  to  rotate  in  the  direction  of  the  arrow,  the 
theoretical  action  begins  at  A  and  ends  at  D,  the  line  AD  being 
termed  the  "line  of  action."  N  It  is  obvious,  however,  that  the 
actual  action  can  only  begin  at  B,  where  the  outside  diameter 
of  the  upper  gear  intersects  the  line  AD,  ending  at  the  corre- 
sponding point  C.  Drawing  involutes  from  these  points  to  the 
base  circle,  and  continuing  the  radials  to  center  O,  the  arc 


STUB-TOOTH   GEARS 


89 


included  between  lines  A  and  D  is  seen  to  be  the  maximum  or 
greatest  possible  arc  of  action,  while  B  and  C  define  the  actual 
arc  of  action. 


Machinery 


Fig.  4.    Analyzing  the  Tooth  Contact  of  the  Gears  in  Fig.  2 


inery 


Fig.  5.    Analyzing  the  Tooth  Contact  of  the  Stub-tooth  Gears  in  Fig.  3 

In  Fig.  4  is  shown  an  involute  curve  of  14!  degrees  obliquity 
from  each  of  the  gears  in  Fig.  2,  the  curves  being  of  sufficient 
length  to  cover  the  maximum  arc  of  action,  and  drawn  to  the 


SPUR   GEARING 


same  scale  as  Figs.  2  and  3.  The  alternately  shaded  divisions 
of  the  curves  show  the  portion  of  each  that  is  in  contact  with  its 
mate  during  an  equal  angular  movement  of  the  gears. 

In  Fig.  5  is  seen  a  similar  diagram  for  Fig.  3,  with  an  angle  of 
obliquity  of  20  degrees. 

In  Figs.  6  and  7  we  have  the  same  involute  curves,  developed 
into  straight  lines,  the  points  corresponding  to  the  divisions  of 
Figs.  4  and  5  being  connected  by  cross  lines. 

To  one  who  has  labored  under  the  impression  that  if  the 
involute  curves  of  a  pair  of  gears  are  correct,  the  action  is  nearly 
a  rolling  one,  a  comparison  of  these  diagrams  will  be  both  in- 


26  TEETH 


Fig.  6.    The  Involute  Curves  in  Fig.  4  developed  into  Straight  Lines 
—  Standard  Involute  Teeth 

teresting  and  instructive.  It  will  be  noted  that  although  the 
divisions  of  the  base  circles  are  equal,  those  of  the  involute 
decrease  as  the  base  is  approached,  showing  that  the  wear  is 
concentrated  at  this  point.  When,  further,  it  is  considered  that 
the  contact  is  between  the  flank  of  this  tooth  and  the  point  of 
its  mate,  it  is  seen  that  the  condition  is  far  from  ideal. 

By  a  comparison  of  Figs.  6  and  7  we  note  that  the  portion  of 
actual  contact,  denoted  by  the  shaded  part,  includes  in  the  case 
of  the  i4|-degree  tooth  cross  lines  that  have  a  considerable 
angularity,  showing  an  excessive  sliding  action;  while  the  corre- 
sponding lines  of  20-degree  teeth  are  nearly  parallel,  denoting 
that  the  action  is  nearly  a  rolling  one. 

Again  referring  to  Figs.  2  and  3  as  showing  a  comparison  of 


STUB-TOOTH   GEARS  91 

the  two  systems,  we  note  the  two  points  in  which  they  differ; 
first,  on  account  of  the  greater  angle  of  the  line  of  action  of  the 
stub- tooth,  the  maximum  arc  of  action  is  much  increased; 
second,  the  ratio  of  the  actual  to  the  maximum  arc  of  action  of 
the  stub-tooth  is  much  less  than  with  the  i4^-degree  tooth. 

This  latter  point  is  a  very  important  one,  as  we  thus  eliminate 
contact  at  both  ends  of  the  line  of  action.  When  we  realize  that 
this  is  the  portion  of  the  action  in  which  the  greater  part  of  the 


Fig.  7.    The  Involute  Curves  in  Fig.  5  developed  into  Straight  Lines 
—  Stub  Teeth 

sliding  takes  place,  with  its  inevitable  wear,  we  see  that  it  is  a 
good  thing  to  cut  out  all  we  can  of  it.  In  the  stub-tooth  the 
point  of  the  tooth  which  wears  out  the  flank  of  its  mate  is  re- 
moved, and  this  reduces  the  friction  while  increasing  the  effi- 
ciency. A  comparison  of  Figs.  6  and  7  shows  that  the  action  of 
the  stub-tooth  is  as  nearly  a  rolling  one  as  it  is  practicable  to 
obtain. 

It  is  of  course  impossible  to  entirely  eliminate  the  wear  between 
the  teeth  of  gears  working  under  a  load.     But  if  the  wear  could 


SPUR   GEARING 


be  evenly  distributed  over  the  entire  working  face  of  the  tooth, 
the  correct  form  of  tooth  would  be  retained  indefinitely  and  a 
worn-out  gear  would,  aside  from  the  excessive  backlash,  run  as 
well  as  a  new  gear.  And  if  this  wear  can  be  evenly  distributed, 
the  durability  of  any  gear  will  be  increased  many  tunes. 

If  the  gear  combinations  18-33  and  14-20  are  analyzed  in  the 
same  manner  as  the  combination  25-26,  it  will  be  found  that  in 
these  cases,  as  in  the  one  first  considered,  the  sliding  action  is 


40  TOOTH   INVOLUTE 
14^°ANGLE  OF  PRESSURE 


20  TOOTH   INVOLUTE 
14^°ANGLE  OF  PRESSURE 


12  TOOTH  STUB 
20°ANGLE  OF  PRESSURE 


12  TOOTH  INVOLUTE 
° ANGLE  OF  PRESSURE 

Machinery 


Fig.  8.    Comparison  between  Standard  and  Stub  Type  of  Gear-teeth 

largely  eliminated  from  the  stub-tooth  gear,  due  to  the  com- 
bination of  the  short  tooth  and  the  increased  pressure  angle. 
It  will  also  be  noted  that  in  the  case  of  the  i^-degree  tooth,  on 
account  of  the  diameter  of  the  base  line  of  the  pinion  approach- 
ing so  closely  that  of  the  pitch  line,  the  entire  length  of  the 
gear  tooth  does  not  have  contact  with  the  pinion.  In  order  to 
accommodate  this  useless  length  of  gear  tooth,  the  tooth  of  the 
pinion  is  materially  and  unnecessarily  weakened.  This  excess 
length  is  about  30  per  cent  of  the  addendum  in  one  case  and 
about  40  per  cent  in  the  other. 


STUB-TOOTH  GEARS  93 

Comparison   between   Ordinary  and   Stub   Gear-teeth.  —  A 

comparison  of  the  regular  standard  gear  tooth  and  the  stub 
gear- tooth,  as  shown  in  Fig.  8,  will  be  of  interest.  Here  are 
shown,  to  the  left,  parts  of  three  stub-tooth  gears,  and  sections 
of  three  standard  i^-degree  involute  gears,  with  teeth  varying 
in  number  from  12  to  40.  The  illustration  gives  a  clear  idea  of 
the  increase  in  strength  with  the  increase  in  pressure  angle. 
The  subject  of  strength  of  stub  gear-teeth  will  be  taken  up  in 
subsequent  pages. 

Summary  of  Conclusions.  —  The  advantages  of  the  stub-tooth 
may  be  stated  as  follows: 

1.  Greater  strength. 

2.  Equal  arc  of  rolling  contact  to  the  i^-degree  involute. 

3.  Extreme  sliding  contact  avoided. 

4.  More  even  wearing  contact. 

As  a  disadvantage,  it  may  be  mentioned  that  we  require  a 
greater  number  of  rotary  cutters  to  cover  a  given  range  of  teeth. 
For  the  standard  involute  system  eight  cutters  are  required, 
whereas  the  stub-tooth  system  requires  sixteen  in  order  to 
produce  good  gears.  When  a  generating  machine,  such  as  the 
Fellows  gear  shaper,  is  used,  only  one  cutter  is  required  for  each 
pitch,  as  the  cutting  action  of  this  machine  is  that  of  a  pinion 
meshing  with  a  gear-wheel. 

The  stub-tooth  system  has  become  thoroughly  established 
in  this  and  foreign  countries.  Since  its  introduction  in  1899  its 
use  has  steadily  increased.  Fully  one-third  of  the  large  cutter 
business  of  the  Fellows  Gear  Shaper  Company  is  for  cutters  of 
the  stub- tooth  form.  Its  use  is  not  now,  as  in  the  beginning, 
confined  to  automobile  work.  It  is  used  in  every  line  of  work 
where  strength,  durability  and  running  qualities  are  demanded. 

Dimensions  of  Stub  Gear-teeth.  —  The  stub  gear-teeth  intro- 
duced by  the  Fellows  Gear  Shaper  Co.  (by  far  the  most  commonly 
used)  are  based  on  the  use  of  two  diametral  pitches.  One  diam- 
etral pitch,  say  8,  is  used  as  the  basis  for  obtaining  the  dimensions 
for  the  addendum  and  dedendum,  while  another  diametral  pitch, 
say  6,  is  used  for  obtaining  the  dimensions  of  the  thickness  of  the 
tooth,  the  number  of  teeth,  and  the  pitch  diameter.  Teeth  made 


94 


SPUR  GEARING 


according  to  this  system  are  designated  as  f  pitch,  jf  pitch,  etc., 
the  numerator  in  this  fraction  indicating  the  pitch  determining 
the  thickness  of  the  tooth  and  the  number  of  teeth,  and  the 
denominator,  the  pitch  determining  the  depth  of  the  tooth.  The 
clearance  is  made  greater  than  in  the  ordinary  gear-tooth  system 
and  equals  0.25-7-  diametral  pitch.  The  pressure  angle  is  20 
degrees. 

Dimensions  of  Stub  Gear-teeth  (Fellows  Gear  Shaper  Co.'s  System) 


Diametral 
Pitch 

Thickness 
of  Tooth 

Addendum 

Working 
Depth 

Depth  of 
Space  below 
Pitch  Line 

Clearance 

Whole 
Depth  of 
Tooth 

% 

0.3927 

O.2OOO 

0.4000 

0.2500 

0.0500 

0.4500 

» 

0.3142 

0.1429 

0.2858 

0.1786 

0-0357 

0.3214 

% 

0.2618 

0.1250 

0.2500 

0.1562 

0.0312 

0.2812 

% 

0.2244 

O.IIII 

O.  2222 

0.1389 

0.0278 

0.2500 

91o 

0.1963 

O.IOOO 

O.2OOO 

0.1250 

0.0250 

0.2250 

9ii 

0-1745 

0.0909 

0.1818 

0.1136 

O.O227 

0.2045 

i91a 

0.1571 

0.0833 

0.1667 

0.1041 

O.O2O8 

0.1875 

iM* 

0.1309 

0.0714 

0.1429 

0.0893 

0.0179 

0.1607 

The  Nuttall  Co.'s  System  of  Stub  Gear-teeth.  —  In  a  system 
of  stub  gear-teeth  originated  by  Mr.  C.  H.  Logue  of  the  R.  D. 
Nuttall  Co.,  the  tooth  dimensions  are  based  directly  upon  the 
circular  pitch.  The  addendum  is  made  equal  to  0.250  X  the 
circular  pitch,  and  the  dedendum  equal  to  0.300  X  the  circular 
pitch.  The  pressure  angle  is  retained  at  20  degrees.  This  sys- 
tem was  adopted  because  of  disadvantages  claimed  to  exist  in 
the  Fellows'  system  using  two  diametral  pitches,  or  one  diam- 
etral pitch  to  obtain  the  addendum  and  dedendum,  and  another 
diametral  pitch  to  obtain  the  thickness  of  the  tooth,  and  the 
number  of  teeth,  etc.  The  chief  disadvantages  of  this  system 
are  that  the  depth  of  the  tooth  is  not  a  direct  function  of  the 
circular  pitch,  and  that  this  system  can  be  used  only  in  combi- 
nation with  diametral  pitch  gears.  When  the  stub  gear-tooth 
is  made  according  to  the  proportions  0.250  and  0.300  times  the 
circular  pitch,  as  just  stated,  the  system  can  be  applied  to  either 
diametral,  circular  or  millimeter  pitch  gears. 

Special  Tooth  Shape  for  Rolling  Mill  Gears.  —  The  illustra- 
tion and  table  below  give  data  for  laying  out  special  22^-degree 


STUB-TOOTH   GEARS 


95 


involute  gears  for  rolling  mill  service.  These  teeth  vary  some- 
what from  the  standard  shape,  and  liberal  fillets  are  provided  at 
the  root  of  the  teeth  to  prevent  breakage  due  to  sharp  corners  at 
these  points.  The  dimensions  given  in  the  table  are  for  i-inch 
circular  pitch.  To  find  the  dimensions  for  any  other  pitch, 
multiply  the  dimension  given  by  the  circular  pitch.  The  length 

Rolling  Mill  Gear-teeth 


_—  -  —  "^^^•X                                                Formulas: 

^t      /5^         ^k                            -4  =  0.2  75  X  circular  pitch; 

V    JC^ET^^-^LPITCH                     5  =  o.325Xcircular  pitch; 

"T^f"              '^pTcmciT--                  C=  0.462  Xpitch  diameter; 

coKIsTrr^Ex^iG  —  ^1    ^  ^                  £  =  0.49  X  circular  pitch  ; 

^"Tl?  D  Hptc^^                  ^=  o-25  1  X  pitch  diameter; 

j^pf                     ^$5?                    G  =o.  136  X  pitch  diameter; 

^                                                     Pressure  angle  =  2  2H  degrees. 

Strength  for 

i"  Pitch, 

No.  of 

Teeth 

Pitch 
Diam. 

Base 
Radius, 
C 

Thick- 
ness, 
D 

Face 
Radius, 
F 

Flank 
Radius, 
G 

Dis- 
tance, 
H 

Dis- 
tance, 

i"  Face, 
Lbs.  at 
Stress  of 

looo  Lbs. 

per  Sq.  In. 

IO 

3.183 

1.470 

0.480 

O.Soo 

0-433 

O.OIO 

O.OIO 

105 

II 

3-501 

1.617 

0.496 

0.879 

0.476 

0.014 

0.014 

"5 

12 

3.820 

1.764 

0.505 

0-959 

0.520 

0.016 

0.016 

117 

13 

4.138 

1  .912 

0.515 

•04 

0-563 

0.005 

0.005 

121 

4.456 

2.059 

0.526 

.12 

0.606 

O.OIO 

0.006 

126 

15 

4-775 

2.206 

0-545 

.20 

0.650 

0.000 

0.000 

134 

16 

5.093 

2-353 

.28 

0.693 

0.005 

0.000 

137 

17 

5-4II 

2.500 

0.570 

.36 

0.736 

0.006 

0.006 

I46 

18 

5-730 

2.647 

0.572 

.44 

0.779 

0.000 

0.000 

19 

6.048 

2.794 

0-575 

•52 

0.823 

0.006 

0.006 

140 

20 

6.366 

2.941 

0.580 

.60 

0.866 

0.014 

0.007 

136 

21 

6.684 

3-088 

0.585 

.68 

0.899 

0.015 

0.000 

136 

of  the  face  of  rolling  mill  pinions  is  made  about  equal  to  the  pitch 
diameter.  The  angle  of  the  tooth  with  a  line  parallel  to  the  axis 
of  the  gear  is  usually  made  30  degrees.  The  figures  given  in  the 
column  headed  "Strength  for  i-inch  Pitch,  i-inch  Face"  are 
based  on  a  working  stress  of  1000  pounds  per  square  inch.  To 
find  the  strength  of  teeth  for  other  dimensions  and  other  working 
stress,  multiply  the  figures  given  by:  face  of  gear  X  circular 

pitch  XW°rkingStreSS. 

IOOO 


96  '  SPUR  GEARING 

Example:  —  Find  allowable  pressure  on  teeth  with  a  working 
stress  of  4000  pounds  per  square  inch,  for  a  2-inch  circular  pitch 
gear  of  4-inch  face  having  20  teeth. 

136  X  4  X  2  X =  4352  pounds. 

1000 

Strength  of  Stub  Gear-teeth.  —  We  have  so  far  discussed  only 
the  points  of  efficiency  and  durability,  but  there  is  another 
advantage  of  the  stub-tooth  over  the  standard  form,  and  one 
which  some  might  think  entitled  to  first  consideration,  especially 
in  the  transmission  of  any  considerable  amounts  of  power;  this 
is  the  advantage  of  greatly  increased  strength. 

Because  of  the  fact  that  the  tooth  has  been  shortened  and  at 
the  same  time  widened  at  its  base,  there  is  a  very  substantial 
gain  in  strength,  which  in  some  cases  amounts  to  about  75  per 
cent.  The  fact  that  we  can  sometimes  almost  double  the  strength 
of  a  tooth  is  certainly  worthy  of  serious  consideration.  In  Fig.  9 
are  shown  enlarged  sections  of  different  combinations  of  gears, 
showing  the  comparative  strength  of  each  form  by  the  well- 
known  graphical  method  of  Wilfred  Lewis. 

In  laying  out  these  diagrams,  we  first  draw  the  normal  of 
the  involute  AB  from  the  extreme  point  of  the  tooth.  From  the 
point  B,  where  the  normal  intersects  the  center  line,  erect  the 
parabola  BC,  with  its  base  tangent  to  the  flank  of  the  tooth  and 
indicating  D  as  the  weakest  section.  Draw  DG  at  right  angles 
to  the  center  line  and  connect  B  and  D.  Then  draw  DH  at  right 
angles  to  BD,  intersecting  the  center  line  at  H.  In  this  construc- 
tion GH  may  be  taken  as  the  measure  of  the  strength  of  the  tooth. 

A  comparison  of  these  diagrams,  of  both  the  standard  and 
stub-tooth  gears,  shows  an  increase  of  strength  in  these  cases 
which  is  very  marked.  It  should  also  be  noted  that  the  gain  in 
strength  is  greatest  in  the  case  of  the  smallest  pinion,  which  is 
a  very  great  advantage;  for  as  a  chain  is  no  stronger  than  its 
weakest  link,  so  a  train  of  gears  is  no  stronger  than  the  teeth 
of  its  smallest  pinion.  This,  in  addition  to  the  clearly  demon- 
strated fact  of  easier  running  and  reduced  wear,  should  command 
careful  consideration  for  the  form  of  tooth  here  advocated. 


STUB-TOOTH   GEARS 


97 


25  TEETH 


14  TEETH 


18  TEETH 


20 c 


Machinery 


Fig.  9.    Illustrations  showing  the  Comparative  Strength  of  Standard 
and  Stub  Gear-teeth 


98 


SPUR   GEARING 


Fig.  10  shows  a  comparison  between  a  pinion  having  15  teeth, 
6  pitch  and  i4j-degree  pressure  angle,  and  one  with  20  teeth,  -j8^ 
pitch  and  stub-tooth  form  of  2o-degree  pressure  angle,  both 
having  the  same  pitch  diameter.  A  comparison  of  the  length  of 
the  line  GH  shows  that  the  stub- tooth,  although  much  shorter, 
has  20  per  cent  greater  strength;  and  that  while  the  bearing  sur- 
face per  tooth  is  shorter,  the  total  area  of  bearing  surface  is  6  per 
cent  greater. 

This  is  not  given  as  a  suggestion  that  the  pitch  be  reduced  to 
secure  an  increase  in  strength  of  gears;  it  merely  shows  one 
of  the  possibilities  of  the  stub-tooth. 


15  TEETH  6  PITCH  20  TEETH   £  PITCH 

ENLARGED  4  TIMES  Machinery 


Fig.  xo.    Comparison  of  Strength  of  Two  Gears  of  Equal  Pitch  Diameter 
and  Unequal  Pitch 

Some  not  entirely  familiar  with  this  system  have  made  the 
mistake  of  thinking  that  it  consists  simply  of  a  shorter  tooth, 
while  retaining  the  same  pressure  angle.  They  have  therefore 
opposed  it  on  the  ground  that  the  arc  of  action  with  a  small 
pinion  would  not  be  equal  to  the  pitch  arc.  The  action  would 
then  not  be  continuous,  because  one  tooth  is  out  of  contact  be- 
fore the  next  tooth  takes  up  the  load.  It  should  be  thoroughly 
understood  that  the  increased  angle  of  obliquity  is  an  essential 
and  vital  part  of  the  stub-tooth  system;  and  that  with  this 
increased  angle,  the  arc  of  involute  action  is  even  longer  than  that 
of  the  14^-degree  standard  tooth. 


STUB-TOOTH   GEARS 


99 


Analysis  of  the  Strength  of  Stub  Gear-teeth.  —  An  investiga- 
tion has  been  made  by  Mr.  L.  L.  Smith  of  the  strength  of  stub 
gear- teeth.  This  analysis  was  published  in  MACHINERY,  January, 
1914.  The  following  pages  contain  an  abstract  of  the  methods 
and  results  of  the  investigation.  Accurate  information  regarding 
the  strength  of  these  teeth  has  been  lacking,  and  it  was  the 
purpose  of  this  investigation  to  determine  correct  values  of  the 
factor  F  in  the  Lewis  formula  for  the  strength  of  gear  teeth  as 
applied  to  this  system  of  gearing.  The  two  forms  of  stub-teeth 
used  in  this  country  —  that 
recommended  by  the  Fel- 
lows Gear  Shaper  Co.,  25 
Pearl  St.,  Springfield,  Vt, 
and  that  recommended  by 
the  R.  D.  Nuttall  Co., 
Pittsburg,  Pa.  —  are  the 
ones  with  which  this  inves- 
tigation is  concerned. 

The  Strength  of  Gearing 

-  Lewis    Assumptions.  — 

Mr.  Wilfred  Lewis  was  the 

first  investigator   to   take 

into  consideration  the  form 

Of  the  tOOth  profile  and  the  'Fig.  „.    Diagram  illustrating  the  Derivation 

fact  that  the  line  of  action  of  the  Lewis  Formula 

of  the  pressure  is  always  perpendicular  to  the  surface.  When 
gears  are  cut  accurately,  his  formula  gives,  as  already  stated, 
very  satisfactory  results,  and  with  a  few  corrections  for  the 
number  of  teeth  in  contact,  etc.,  it  is  almost  universally  used 
in  gear  design. 

Mr.  Lewis  assumed  that  all  of  the  load  on  the  gear  was  con- 
centrated at  the  end  of  one  tooth,  its  line  of  action  being  perpen- 
dicular to  the  surface  AC  as  shown  in  Fig.  n.  (See  also  Chapter 
III  for  the  derivation  of  factor  Y  in  the  Lewis  formula.)  The 
actual  force  P  was  resolved  into  a  tangential  force  W  and  a 
radial  force  Q.  The  tangential  force  produces  a  bending  stress 
in  the  tooth,  while  the  radial  force  produces  a  uniformly  dis- 


Maohinery 


100  SPUR   GEARING 

tributed  compressive  stress.  When  the  material  of  which  the 
gears  are  made  is  stronger  in  compression  than  in  tension,  this 
radial  component  is  a  source  of  strength,  but  when  the  material 
is  of  about  the  same  strength  in  tension  and  compression  it  is 
a  source  of  weakness.  For  small  angles  of  obliquity,  a,  this 
compression  does  not  amount  to  more  than  10  per  cent  of  the 
total  stress  and  on  this  account  was  neglected  by  Mr.  Lewis. 
The  tangential  force  W  he  assumed  to  be  equal  to  the  force 
transmitted  at  the  pitch  circle.  This  last  assumption,  although 
perhaps  as  much  as  10  per  cent  in  error  for  small  pinions,  gives 
values  of  the  stress  on  the  safe  side  and  so  may  be  considered 
accurate  enough  for  practical  purposes. 

Force  Analysis.  —  To  obtain  the  weakest  section  of  the  tooth, 
Mr.  Lewis  constructed  a  parabola  CBF  (Fig.  n)  passing  through 
B,  the  application  point  of  the  tangential  force  W,  and  tangent 
to  the  tooth  profile  at  C  and  F.  This  parabola  encloses  a  canti- 
lever beam  of  uniform  strength  and  it  can  be  readily  seen  that 
the  weakest  section  of  the  tooth  lies  along  CF.  The  resisting 
moment  of  this  section  is: 

M-f   ........,„ 

where  5  =  allowable  fiber  stress; 

/  =  face  width  of  the  gear. 

The  bending  moment  is  Wh,  hence: 


Wh  =  (2) 

6 

From  the  geometry  of  the  figure: 

*=£:    .....   •   ••  •   (3) 

4# 


Then  W  =  =  Sfp'  X       , 

where  pr  =  the  circular  pitch. 

Representing  the  ratio  -^7  by  the  symbol  F,  we  have: 
3P 

W  =  Sp'fY     .......     (4) 


STUB-TOOTH   GEARS  IOI 

The  value  F  is  thus  seen  to  be  dependent  upon  the  tooth 
profile  and  the  pitch  of  the  gear.  By  drawing  the  tooth  pro- 
files for  gears  with  various  numbers  of  teeth,  Mr.  Lewis  was  able 
to  calculate  the  value  of  F,  for  the  standard  involute  system, 
and  found  that  the  following  equation  very  closely  fitted  the 
values  he  obtained. 


in  which  Yr  =  the  value  of  F  for  rack  teeth; 

a  =  constant  depending  upon  angle  of  obliquity; 
n  =  number  of  teeth  in  gear. 

A  mathematical  analysis  of  a  rack  tooth  outline  shows  that 
the  value  of  Fr  is  given  by  the  following  equation: 

....     (6) 


in  which  /  =  thickness  of  tooth  at  pitch  line; 

z  =  addendum; 
p  =  diametral  pitch. 

Stub-tooth  Systems.  —  To  meet  the  demand  for  gear  teeth 
that  are.  stronger  than  the  standard  i4§-degree  involute,  various 
methods  of  increasing  their  strength  may  be  resorted  to.  Two 
of  these  methods  are  as  follows:  (i)  shortening  the  addendum 
and  keeping  the  standard  angle  of  obliquity;  and  (2)  increasing 
the  angle  of  obliquity  and  preserving  the  standard  height  of  the 
tooth.  The  stub-tooth  system  is  a  combination  of  these  two 
methods,  in  which  the  height  of  the  tooth  is  decreased  to  about 
0.8  the  standard  height  and  the  angle  of  obliquity  increased  to 
20  degrees. 

Advantages  of  Stub-tooth  Systems.  —  The  chief  advantage  of 
this  system  is  by  many  held  to  be  that  the  teeth  will  transmit 
greater  loads,  and  this  fact  is  taken  advantage  of  in  the  con- 
struction of  machine  tools,  hoisting  machinery  and  automobiles, 
because  with  the  same  quality  and  amount  of  material,  greater 
strength  can  be  obtained.  The  stub-tooth  gears  run  smoother 
than  gears  with  the  standard  i^-degree  involute  teeth,  due  to 
the  decreased  impact,  because  for  a  given  strength  and  length 


102 


SPUR  GEARING 


of  face,  the  pitch  is  smaller  and  consequently  the  number  of 
teeth  in  mesh  is  greater.  Stub- tooth  gears  are  almost  universally 
used  by  automobile  manufacturers,  and  they  report  that  the 
action  is  smoother  and  the  wear  less  than  with  standard  gears. 
Mr.  Norman  Litchfield  in  the  Transactions  of  the  American 
Society  of  Mechanical  Engineers  for  1908  reports  that  they  have 
given  excellent  service  on  the  New  York  subway  trains. 

Mr.  E.  R.  Fellows  in  the  same  volume  of  the  Transactions 
presents  the  chart  shown  in  Fig.  12,  which  is  based  on  actual 


10 


30        40         50         60         70 
SHOP  EFFICIENCY-PER  CENT 


80 


90      100 


Machinery 


Fig.  la.    Diagram  showing  Relative  Efficiency  of  Stub  and  Standard 
Gear  Teeth 

experience.  The  curves  apply  to  stub-tooth  and  standard  gears 
with  "shop  efficiency"  plotted  against  "running  quality."  By 
shop  efficiency  is  meant  the  relative  conditions  under  which  the 
gears  operate  and  that  100  per  cent  would  only  be  attained  under 
laboratory  conditions,  90  per  cent  representing  first-class  com- 
mercial conditions  and  70  per  cent  the  common  conditions  in 
an  average  shop.  By  running  quality  is  meant  the  relative 
noiselessness  of  the  gears,  100  per  cent  signifying  absolutely 
noiseless  operation  and  90  per  cent  the  best  condition  actually 


STUB-TOOTH   GEARS 


103 


obtainable.  The  curves  show  that  under  the  very  best  condi- 
tions the  two  gears  operate  about  equally  well,  but  under  average 
commercial  conditions  the  stub-tooth  gear  is  decidedly  superior. 
Determination  of  the  Strength  Factor  and  Method  of  Drawing 
Profiles.  —  In  order  to  determine  the  value  of  7  for  a  certain 
gear  of  a  given  pitch,  it  is  necessary  to  lay  out  a  tooth  to  some 
magnified  scale  in  the  same  manner  (illustrated  by  Fig.  n)  that 
Mr.  Lewis  measured  the  distance  x  and  computed  Y.  In  this 
investigation,  this  has  been  done  for  a  great  many  pitches,  using 
a  scale  of  10  to  i,  except  for  the  large  pitches  where  the  drawings 


CUTTER  PROFILE 


Machitiery 


Fig.  13.    Method  of  laying  out  Gear  Teeth  to  Determine  the  Values 
of  jsrand  Y 

would  become  too  large.  For  pitches  f  and  f ,  a  scale  of  8  to  i 
was  used  and  for  -|  pitch  a  scale  of  6  to  i.  Fig.  13  illustrates  the 
method  of  laying  out  the  teeth.  Before  laying  out  the  teeth  the 
profile  of  the  cutter  tooth  was  drawn  on  tracing  paper  with 
the  addendum  equal  to  the  dedendum  of  the  stub-tooth  gear,  and 
the  diameter  equal  to  that  of  a  24-tooth  gear.  The  tooth  profiles 
of  gears  with  various  numbers  of  teeth  were  then  drawn  accu- 
rately on  tracing  paper,  with  the  face,  from  the  base  circle  out,  a 
true  involute  and  the  flank  and  fillet  generated  by  the  point  of 
the  cutter  tooth,  as  the  pitch  circles  of  the  gear  and  cutter  were 
rolled  together. 


IO4 


SPUR   GEARING 


Having  thus  obtained  the  profile,  the  action  line  AB  (Fig.  13) 
of  the  force  at  the  end  of  the  tooth  was  drawn  normal  to  the  in- 
volute, i.e.,  at  20  degrees  to  the  vertical.  In  order  to  get  the 
parabola  BCF  passing  through  B  and  tangent  to  the  profile,  a 
considerable  number  of  parabolas,  differing  but  slightly  from 
each  other,  were  drawn  on  cards,  so  that  they  could  be  slipped 
in  under  the  tracing  paper  and  adjusted  to  enable  the  point  of 
tangency  C  to  be  determined.  The  line  BC  was  then  drawn  and 
CD  and  CE  drawn  perpendicular  to  BC  and  BD,  respectively. 
The  length  x  was  measured  as  accurately  as  possible  with  a 
Starrett  steel1  scale  and  the  value  of  Y  computed.  Two  or  more 


ALL  STAND 


NUTTALL  STANDARD 

12  T    6  PITCH        Machinery 


Fig.  14.    Tooth  Profiles  and  Values  of  x  for  Different  Pitches 

profiles  of  the  same  tooth  were  sometimes  drawn  and  it  was  found 
that  the  values  of  x  usually  checked  within  i  or  2  per  cent.  Fig. 
14  shows  the  tooth  profiles  and  values  of  x  for  various  pitches. 
Method  of  Plotting.  —  We  have  already  determined  an  exact 
method  of  computing  Yr  in  Equation  (5)  : 


n 


If  any  method  could  be  found  for  obtaining  the  value  of  -  , 
the  problem  of  determining  the  value  of  Y  would  be  very  simple. 


STUB-TOOTH   GEARS 


105 


The  use  of  logarithmic  cross-section  paper  makes  the  plotting 
of  all  such  functions  as  F  =  bxa  very  simple,  because  the  curve 

becomes  a  straight  line  with  a  slope  a.     In  Equation  (5)  -  is  a 

n 

function  of  the  same  character,  and  when  so  plotted  for  different 
values  of  n,  it  should  give  a  straight  line  with  a  slope  of  —  i. 
Advantage  was  taken  of  this  peculiar  property  of  logarithmic 
paper  by  first  carefully  computing  F  and  Fr  for  five  different 
gears  for  each  pitch,  and  then  plotting  the  corresponding  values 

of  - .  Due  to  unavoidable  errors  in  calculating  F,  the  points  did 
not  all  lie  exactly  in  a  straight  line,  but  somewhat  as  shown  in 
Fig.  15,  which  gives  values  of  ~  for  the  f  pitch  teeth,  and  so  it 

was  considered  best  to  use  five  points  and  draw  an  average  line 
through  them  instead  of  drawing  a  line  through  just  two  deter- 
mined points. 

It  was  found  that  instead  of  the  function  being  -  it  was  — 

n  nm 

for  stub- tooth  gears  with  the  slope  of  the  line,  i.e.,  the  value  of 
m,  lying  between  0.72  and  0.93.  Values  of  a  and  m  are  given  in 
the  table  below. 

Table  of  Values  of  a  and  m 


Stand- 

Pitch 

n 

& 

N 

X 

Ha 

Hi 

»H« 

»M4 

Nuttall 
System 

ard 

20-deg. 
In- 

volute 

a 

m 

0-505 
0.760 

0-535 
0.800 

0.605 
0.830 

0.760 
0.930 

0.690 
0.890 

0.490 
0.780 

0.540 
0.800 

0.440 
0.720 

0.592 
0.820 

0.540 
0.790 

The  values  of  F  in  the  table  on  the  next  page  were  then  ob- 
tained by  simply  subtracting  the  value  of  —  obtained  by  means 

of  the  proper  chart  from  Fr.  If  this  method  of  plotting  had  not 
been  used,  it  would  have  been  necessary  to  lay  out  gears  with 
from  1 2  to  200  teeth  and  measure  x  on  each  drawing,  a  task  which 
would  have  taken  five  or  six  times  as  long  to  do. 


io6 


SPUR   GEARING 


Conclusions.  —  One  of  the  interesting  facts  that  was  brought 
out  in  this  investigation  was  that  the  function  —  as  given  by 

Lewis  does  not  apply  to  stub-tooth  gears,  and  that  a  function  of 
this  nature  to  be  correct  should  have  an  exponent  other  than 

Values  of  Y  in  Lewis  Formula  for  Stub-tooth  Gears  (£lrcuUr 


No.  of 
Teeth 

Fellows  System 

Nuttall 
System 

% 

Vi 

% 

% 

Ha 

%  i 

10/12 

»H« 

12 

0.096 

O.III 

O.IO2 

O.IOO 

0.096 

O.IOO 

0.093 

0.092 

0.099 

13 

O.IOI 

0.115 

O.IO7 

0.106 

O.IOI 

0.104 

0.098 

0.096 

0.103 

14 

0.105 

0.119 

O.  112 

O.III 

0.106 

0.108 

0.102 

O.IOO 

0.108 

15 

0.108 

0.123 

O.II5 

0.115 

O.IIO 

O.III 

0.105 

0.103 

O.III 

16 

O.III 

0.126 

O.II9 

0.118 

0.113 

0.114 

0.109 

0.106 

0.115 

I? 

0.114 

0.129 

O.I22 

O.I2I 

0.116 

0.116 

O.III 

0.109 

0.117 

18 

0.117 

0.131 

O.I24 

0.124 

0.119 

0.119 

0.114 

O.III 

O.I  2O 

!Q 

0.119 

o.i33 

O.I27 

0.127 

0.122 

O.I2I 

0.116 

0.113 

0.123 

20 

O.I2I 

o.i35 

0.129 

0.129 

0.124 

0.123 

0.118 

0.115 

0.125 

21 

0.123 

o.i37 

O.I3I 

0.131 

O.I26 

0.125 

O.I  2O 

0.117 

0.127 

22 

O.I25 

0.139 

0.133 

0.133 

0.128 

0.126 

O.I22 

0.118 

0.128 

23 

O.I26 

0.141 

0.134 

0.135 

O.I29 

0.128 

0.123 

O.I  2O 

0.130 

24 

0.128 

0,142 

0.136 

0.136 

O.I3I 

0.129 

0.125 

O.I2I 

0.131 

25 

O.I29 

0.143 

0.137 

0.138 

0.133 

0.130 

0.126 

0.123 

0.133 

26 

0.130 

0.145 

0.139 

0.139 

0.134 

0.132 

0.128 

0.124 

0.134 

27 

0.132 

0.146 

O.I4O 

0.140 

0.135 

0.133 

0.129 

0.125 

0.136 

28 

0-133 

o.i47 

O.I4I 

0.141 

0.136 

0.134 

0.130 

0.126 

0.137 

29 

0.134 

0.148 

O.I42 

0.143 

0.137 

0.135 

0.131 

0.127 

0.138 

30 

0.135 

0.149 

0.143 

0.144 

0.138 

0.136 

0.132 

0.128 

0.139 

32 

0.137 

0.150 

0.145 

0.146 

O.I4O 

0.137 

0.134 

0.130 

0.141 

35 

0.139 

o.i53 

0.147 

0.148 

0.143 

0.139 

0.136 

0.132 

0.143 

37 

0.140 

0.154 

0.149 

0.149 

0.144 

0.141 

0.138 

0.133 

0.145 

40 

O.I42 

0.156 

O.I5I 

0.151 

0.146 

0.142 

0.140 

0.135 

0.146 

45 

0.145 

0.159 

0.154 

0.154 

0.149 

0.145 

0.142 

0.138 

0.149 

50 

0.147 

0.161 

0.156 

0.156 

O.I5I 

0.147 

0.144 

o.  140 

0.151 

55 

0.149 

0.162 

0.157 

0.158 

0.152 

0.149 

o.  146 

0.141 

0.153 

60 

0.150 

0.164 

0.159 

0.159 

0.154 

0.150 

0.148 

0.143 

0.154 

70 

0.153 

0.166 

0.161 

0.161 

0.156 

0.152 

0.150 

0.145 

0.157 

80 

0.155 

0.168 

0.163 

0.163 

0.158 

0.154 

0.152 

0.147 

0.159 

100 

0.158 

o.  171 

0.166 

o.  166 

0.160 

0.156 

0.154 

0.150 

0.161 

150 

O.l62 

0.174 

0.170 

o.  169 

0.164 

o.  160 

0.158 

0.154 

0.165 

200 

O.l64 

0.176 

o.  172 

o.  171 

o.  1  66 

o.  162 

o.  160 

0.156 

0.167 

Rack 

0.173 

0.184 

0.179 

o.  176 

o.  172 

0.170 

0.168 

0.166 

0.175 

I 

unity  in  the  denominator.     In  view  of  this  discovery,  it  was 
thought  strange  that  for  standard  teeth  this  exponent  should  be 

exactly  unity  and  the  function  be  -,  or  as  Lewis  gives  it  for 

20-degree  involute  teeth  °'912 .     Consequently  Lewis  values  of 

n 


STUB-TOOTH  GEARS 


107 


- ,  derived  from  his  values  of  F,  were  plotted  and  it  was  found 


n 


0.012  . 
— - —  is  so 


that  the  function  really  is          ;  but  as  the  function 

tfl   '  iv 

much  simpler  to  calculate,  and  is  correct  within  5  per  cent,  one 
might  be  justified  in  using  it. 


0.10 
0.09 
0.08 

0.07 
0.06 

0.05 
0.04 

4 

JL 

co  0.03 

3 

0.02 
0.01 

5jT 

X, 

x 

^v 

\ 

Sj 

\ 

\ 

\ 

X 

s 

\ 

\ 

\ 

s 

\ 

«/8  PITCH 

yr=0.1 

79 

\ 

\ 

\ 

s 

\ 

S 

s 

\ 

s 

0                                      20                      30             40           50        60       70     80    90    100 
NUMBER  OF  TEETH                                                 Machinery 

Fig.  15.    Diagram  showing  the  Values  from  which  Factor  Y  is  Derived 
for  a  %  Pitch  Gear 

Owing  to  the  varying  ratio  of  the  addendum  to  the  circular 
pitch  in  the  Fellows  system,  the  values  of  Y  are  different  for 
each  pitch,  but  in  the  Nuttall  system  this  ratio  is  constant,  Y 
being  the  same  for  all  pitches.  The  values  of  Y  in  the  table 
indicate  that  stub- tooth  pinions  with  less  than  25  teeth  are  about 
25  per  cent  stronger  than  the  standard  2O-degree  involute  and 
40  per  cent  stronger  than  the  standard  i^-degree  involute.  For 
larger  gears  the  difference  is  in  favor  of  the  stub-tooth,  but  is  not 
quite  so  marked. 


CHAPTER  VI 
NOISY   GEARING 

A  GREAT  deal  has  been  written  on  the  subject  of  noisy  gearing. 
Many  suggestions  have  been  made  for  its  elimination  and  some 
improvements  have  undoubtedly  been  made.  Rawhide  and 
cloth  gears,  referred  to  in  previous  chapters,  meet  the  require- 
ments in  many  instances;  but  often  metal  to  metal  gearing  is 
required  by  the  conditions  of  design  or  application.  In  the 
present  chapter  a  method  will  be  explained  which  tends  to  pro- 
duce silent-running  metal  gearing.  This  method  has  been 
practically  applied  for  several  years. 

Causes  of  Noisy  Gearing.  —  All  noise  in  gearing  is  caused  by 
the  vibration  of  the  material  in  the  gear.  The  source  of  this 
vibration  is  usually  a  series  of  blows  resulting  from  one  or  more 
of  the  following  causes: 

1.  —  The  individual  teeth  are  unequally  loaded;   that  is,  the 
load  is  borne  by  more  teeth  at  some  periods  than  at  others.    In 
this  case  it  is  perhaps  incorrect  to  describe  the  effect  on  the  teeth 
as  a  blow,  since  it  takes  the  form  of  an  increased  compression 
of  the  material,  which,  however,  produces  vibration. 

2.  —  The  blow  may  be  an  actual  concussion  caused  by  one 
tooth  being  disengaged  before  the  next  tooth  takes  up  the  load. 
Under  present  conditions  this  is  a  cause  seldom  met  with,  but 
was  often  found  in  the  past  when  pinions  with  too  few  teeth 
were  used. 

3.  —  Owing  to  inaccuracies  in  cutting,  the  pitch  of  the  teeth 
may  vary  around  the  gear  circumference,  so  that  while  theoret- 
ically two  or  more  teeth  should  be  in  contact,  only  one  supports 
the  load.     When  the  latter  comes  out  of  contact,  the  load  is 
transmitted  to  the  next  tooth  by  a  sharp  blow.     This  has  been 
a  most  prolific  cause  of  noisy  gearing. 

108 


NOISY  GEARING 


109 


4.  —  The  faulty  alignment  of  the  shafts  on  which  the  gears 
are  mounted  produces  a  jamming  action,  causing  an  objection- 
able grinding  noise.  Even  if  the  alignment  is  perfect  when 
the  gears  are  erected,  the  shafts  are  practically  certain  to  become 
displaced  sooner  or  later  owing  to  uneven  wear  in  the  bearings. 
Figs,  i  and  2  show,  diagrammatically,  a  case  of  this  kind.  In 
Fig.  i  the  power  is  transmitted  from  pinion  A  to  gear  B.  As- 


Machinery 


Figs,  i  and  2.    Diagrammatical  View  showing  Effect  of  Disalignment 

sume  that  the  action  between  the  teeth  is  perfect  and  that, 
therefore,  the  pressure  between  the  teeth  is  at  all  times  in  exactly 
the  same  direction  relative  to  the  pitch  circles.  This  pressure 
tends  to  force  B  in  the  direction  of  the  lower  horizontal  arrow 
and  A  in  the  opposite  direction.  Actual  motion  is  prevented 
by  bearings  C  placed  as  close  to  the  gears  as  possible,  but,  in 
time,  the  pressure  on  the  bearings  C  will  cause  wear;  if  the 
bearing  surface  is  inadequate,  which  is  commonly  the  case, 


HO  SPUR   GEARING 

perceptible  wear  soon  takes  place.  The  result  is  that  the  shafts 
D  tend  to  turn  bodily  about  their  bearings  at  E  where  the  wear 
is  not  likely  to  be  so  pronounced.  The  gears  then  take  a  position 
as  indicated  in  Fig.  2  and  the  teeth  will  bear  on  both  sides. 
This  is  undoubtedly  the  reason  why  gears  originally  silent  in 
action,  gradually  become  noisy.  This  defect  is  difficult  to  cure 
entirely,  but  a  great  deal  may  be  done  by  increasing  the  available 
bearing  surface  close  to  the  gears,  or  by  placing  another  pair  of 
bearings  outside  of  the  gears.  Of  course,  the  forces  acting  on 
the  bearings  are  not  horizontal,  but  slightly  inclined,  owing  to 
the  inclined  action  on  the  teeth.  For  simplicity  of  illustration, 
however,  the  case  has  been  presented  as  indicated. 

5. — Noises  are  also  due  to  special  forms  of  teeth.  For 
example,  with  cycloidal  teeth,  the  smallest  separation  of  the 
shafts  destroys  the  uniformity  of  transmission  and  noise  results. 
Since,  however,  the  involute  tooth  is  now  practically  universally 
adopted,  the  present  chapter  will  deal  with  the  latter  form  of 
tooth  only.  The  statement  that  a  partial  separation  of  the  cen- 
ters of  involute  gears  does  not  affect  the  true  working  of  the  teeth 
is  not  wholly  true;  cases  may  occur  when  noise  will  result  from 
this  cause.  A  separation  of  the  centers  has  the  effect  of  reducing 
the  length  of  time  of  contact,  and  hence  it  is  reasonable  to  assume 
a  case  when  two  gears  have  such  a  number  of  teeth  that  one 
tooth  is  released  from  the  load  at  the  instant  when  the  next  tooth 
in  succession  comes  into  mesh.  In  this  case  the  conditions 
mentioned  under  (2)  are  met  with. 

6.  —  Interference  between  the  teeth  themselves  is  a  common 
cause  of  noisy  gearing.  To  rightly  understand  this  cause  it 
is  necessary  to  enter  briefly  into  the  theory  of  the  shape  of 
the  involute  tooth.  The  involute  is  commonly  denned  as  a 
curve  described  by  the  end  of  a  string  as  it  is  unwound  from 
a  cylinder,  the  string  being  kept  taut,  so  that  in  every  posi- 
tion it  may  be  described  as  a  tangent  to  the  cylinder.  In  Fig.  3, 
A  represents  the  cylinder  and  B  the  string  in  various  positions 
as  it  is  unwound  from  the  periphery  of  A]  C  is  the  involute 
described  by  the  end  of  the  string.  The  circle  A  is  known  as 
the  "base  circle"  of  the  involute,  and  D  is  called  the  "source  of 


NOISY   GEARING 


III 


the  curve."  It  follows  from  the  definition  that  the  involute  is 
not  a  closed  curve;  in  other  words,  it  terminates  in  infinity. 
It  is  also  evident  that  it  is  a  curve  of  two  branches  because  the 
string  may  obviously  be  unwound  from  the  "base  circle"  in 
either  a  clockwise  or  a  counter-clockwise  direction,  the  second 
branch  being  indicated  in  Fig.  3  by  the  dotted  curve  E.  As 
the  involute  lies  entirely  without  the  circumference  of  its  base 
circle,  the  working  part  of  the  curve  terminates  at  D.  It  is, 
therefore,  evident  that  we  must  so  proportion  the  mating  in- 
volutes that  when  the  two  curves  are  rolling  together,  the  point 
of  contact  between  the  source  of  one  curve  with  the  other  curve 
shall  be  the  outer  termination  of  the  latter. 


P10.3 


&S-4  Machinery 


Figs.  3  and  4.    The  Involute  Curve  and  Its  Application  to  Gear  Teeth 

This  will  be  made  clearer  by  referring  to  Fig.  4.  Here  A 
and  B  are  two  base  circles  with  their  respective  involutes  E 
and  F;  D  is  the  source  of  E,  and  C  of  F.  The  two  involutes 
are  in  contact,  and  it  will  be  seen  at  once  that  the  involute  E 
must  be  cut  off  at  the  point  which  has  been  in  contact  with  C, 
and  similarly  that  F  must  terminate  at  the  point  which  will 
come  ultimately  into  contact  with  D.  If  the  involutes  are 
extended  beyond  these  points,  the  mathematical  action  still 
continues  but  actual  contact  is  impossible,  since  it  entails  an 
overlapping  of  the  curves,  causing  one  tooth  to  dig  into  the 
other.  The  points  C  and  D  are  known  as  "  interference  points," 
and  interference  or  the  digging  of  one  tooth  into  its  mate  is  a 


112  SPUR   GEARING 

direct  result  of  extending  the  addendum  beyond  the  circle  drawn 
through  the  interference  point. 

Noise  Due  to  Interference.  —  Evidence  that  interference 
actually  occurs  in  practice  will  usually  be  found  on  examining  a 
pair  of  gears  which  have  been  in  action  for  some  time  and  where 
one  of  them  is  a  pinion  of  less  than  20  teeth.  On  examining  the 
pinion  teeth,  a  groove  will  be  found  in  the  face  of  the  teeth, 
slightly  below  the  imaginary  pitch  line  of  the  tooth,  and  the 
points  of  the  gear  teeth  will  be  found  to  be  rounded  over  and 
bright.  In  some  cases  this  groove  becomes  a  keen  line,  as  if  it 
had  been  drawn  with  a  scriber.  Except  in  the  case  of  equal 
gears,  the  groove  is  usually  found  only  on  the  teeth  of  the  gear 
with  the  lesser  number  of  teeth.  The  presence  of  this  mark  is 
invariably  accompanied  by  noise,  and  is  a  sure  sign  of  there  being 
too  few  teeth  in  the  pinion.  It  is  proposed  to  deal  more  fully 
with  this  point  later,  since  it  deserves  a  great  deal  more  attention 
than  it  generally  receives.  The  blow  which  causes  vibration 
and  noise  in  this  case  takes  place  between  the  points  of  the  gear 
teeth  and  the  flanks  of  the  pinion  teeth.  On  first  contact,  it  is 
a  blow  pure  and  simple,  afterward  becoming  an  abrasion,  so 
that  from  this  cause  two  distinct  kinds  of  noises  arise,  namely, 
the  ring  due  to  the  blow  and  the  grind  due  to  the  abrasion. 
Noisy  gears  must  of  necessity  be  inefficient  transmitters  of  power. 
Therefore,  altogether  apart  from  the  question  of  comfort,  noise 
must  be  reduced  as  far  as  possible  if  high  efficiency  is  to  be 
attained. 

The  foregoing  paragraphs  give  the  six  chief  causes  of  noise 
in  gearing.  In  the  cutting  of  gearing  by  the  various  processes 
now  in  use,  the  limit  of  refinement  in  workmanship  has  prac- 
tically been  reached.  It  is  therefore  evident  that  in  order  to 
eliminate  the  noise  of  gearing,  it  is  necessary  to  make  a  change 
either  in  the  design  of  the  teeth  or  in  the  gears  themselves. 
One  method  which  has  been  adopted  for  the  reduction  of  noise 
is  the  use  of  pinions  made  of  rawhide,  paper,  fiber  and  other 
similar  materials.  Gears  made  from  these  materials,  however, 
do  not  remove  the  cause  of  the  trouble,  but  merely  allay  it,  and 
the  very  causes  which  produce  the  noise  in  regular  gearing  in 


NOISY  GEARING  113 

most  cases  end  by  destroying  the  pinion  when  made  from  less 
wear-resisting  materials.  The  materials  mentioned,  by  their 
very  nature,  do  not  always  promise  good  wearing  resistance  — 
with  the  exception  probably  of  fiber.  Besides,  other  troubles 
are  introduced:  rawhide  is  easily  injured  by  oil,  and  the  same 
applies  to  paper  pinions;  consequently  lubrication  is  difficult,  it 
becoming  necessary  to  use  solid  lubricants,  which  are  somewhat 
difficult  to  apply.  Fiber  is  peculiarly  susceptible  to  moisture, 
which  causes  it  to  swell  and  often  to  jam.  These  materials  for 
gearing  cannot,  therefore,  be  considered  a  permanent  aid  for 
securing  efficiency  of  transmission. 

Analysis  of  the  Conditions.  —  It  has  already  been  shown  that 
for  any  two  gears  in  mesh,  owing  to  the  nature  of  the  involute 
curve,  interference  points  exist,  and  that  if  circles  are  drawn 
through  these  points,  the  points  of  the  teeth  must  not  be  pro- 
longed beyond  these  circles.  This  being  so,  we  are  led  to  believe 
that,  given  a  ratio,  there  should  be  one  pair  of  diameters  which 
will  give  the  best  results,  and  one  pair  only.  It  is  proposed  to 
prove  that  such  is  actually  the  case,  and  to  proceed  to  establish 
formulas  which  will  enable  us  to  fix  the  correct  number  of  teeth 
in  the  larger  gear  of  a  pair  for  any  ratio.  The  present  chapter 
has  been  written  particularly  with  the  object  of  bringing  this 
important  matter  before  the  practical  man. 

In  Fig.  5  is  shown,  diagrammatically,  a  pair  of  gears,  and 
the  number  of  teeth  has  been  purposely  chosen  very  low  in 
order  to  render  the  argument  more  obvious.  The  tooth  pro- 
files are  assumed  to  be  involutes  and  hence  the  line  of  contact 
is  a  straight  line  AB.  The  base  circle  (that  is  the  circle  from 
which  each  involute  is  generated)  is  marked  for  each  gear. 
Now,  the  limit  line  for  each  gear  is  found  by  striking  from  the 
gear  center  an  arc  passing  through  the  point  of  tangency  of  the 
line  of  contact  AB  and  the  base  circle  of  the  other  gear  (i.e,, 
the  source  of  the  working  Involute) .  These  points  are  D  and  E, 
respectively,  and  the  limit  circles  are  shown  passing  through 
them.  It  will  at  once  be  seen  that,  as  far  as  the  pinion  is  con- 
cerned, no  interference  is  to  be  anticipated,  since  the  addendum 
circle  lies  well  within  the  limit  circle,  but  a  considerable  shorten- 


SPUR   GEARING 


ing  of  the  addendum  of  the  gear  teeth  is  necessary  if  interference 
is  to  be  avoided.  The  addendum  is  usually  taken  as  a  function 
of  the  pitch,  the  value  most  commonly  used  being : 


Addendum  = 


diametral  pitch 


Machinery 


Fig.  5.    Diagrammatical  Lay-out  for  the  Derivation  of  Formulas 

It  will,  therefore,  be  seen  that  if  standard  teeth  are  to  be 
used,  it  is  necessary  to  redesign  the  gears,  adjusting  the  num- 
ber of  teeth  and  pitch  so  that  the  addendum  circle  and  limit 
circle  shall  at  least  coincide.  The  best  conditions  are  secured 
when  these  two  circles  coincide  because  the  maximum  arc  of 
contact  without  interference  is  then  obtained.  Further,  it  is 


NOISY  GEARING  115 

obvious  that  since  the  height  of  the  teeth  in  the  larger  gear 
is  dependent  on  the  point  of  tangency  of  the  line  of  contact 
and  the  base  circle  of  the  pinion,  any  variation  in  the  ratio  of 
the  train  by  altering  the  pitch  diameter  of  the  pinion,  and 
consequently  also  its  base  circle  diameter,  will  entail  an  altera- 
tion in  the  tooth  height  for  the  gear;  in  other  words,  there  is 
for  every  ratio  one  pair  of  gears,  and  one  only,  which  will  give 
the  best  all  around  efficiency. 

The  requirements  are  filled  when  the  first  point  of  the  con- 
tact of  the  gear  (the  intersection  between  the  line  of  contact  AB 
and  the  addendum  circle)  is  so  located  that  a  radius  from  the 
center  of  the  pinion,  through  this  point,  makes  a  right  angle  with 
the  line  of  contact.  In  that  case,  the  addendum  circle  and  the 
limit  circle  of  the  gear  will  coincide. 

In  Fig.  5  let  angle  SDO  be  a  right  angle,  and  let  the  angle 
SDC  of  the  involute  be  called  0.  The  line  CD  is  perpendicular 
to  line  MO,  the  line  through  the  gear  centers.  Then,  angle 
SOD  =  angle  SDC  =  </>. 

Let  SD  =  a,     and    SC  =  x.    Then,  -  =  sin  <f>.     .     .     .     (i) 

a 

Let  n  =  number  of  teeth  in  pinion; 
N  =  number  of  teeth  in  gear; 
P  =  diametral  pitch. 

mi_ 

Then, 

Hence,  a  =  —  sin  <£  ......  '  ;.    .    »    •    •     (2) 

Substituting  in  (i)  and  transposing: 

*-^sin*«.    .    .    .    .    .    .    .    (3) 

CD 

In  triangle  CDO  we  have  —  -  =  tan  <f>. 


But,         CO  =  SO  -  SC  =  -2   ~ 


Il6  SPUR  GEARING 

Hence, 


en 

Let  angle  CMD  =  <*;     then,  tan  a 


CM     SM  +  SC 

A7 

But 


Hence>  tan«-  ^  (5) 


2P  '  2P 
Dividing  (4)  by  (5) : 

tan  <f>      TV  +  w  sin2  <^>  TV 


tan  a         n  cos2  <£         w  cos 


2  2 


+ 


Hence,  cot  a  =  —  7—          -  +  tan  <£  .....     (6) 

n  sin  0  cos  <j> 

from  which  a  can  be  determined. 

We  have  further,  a  +  0  +  7  =  1  80  degrees.  But  7  =  90°  +  <£. 

Therefore,  0  =  90°  -  0  -  a.      .     ...    *    •     •     (?) 

from  which  )8  can  now  be  found.     Further, 

DM       SM 

sin  7      sin  (8 

TV  4-  2 


But  DM  =  half  the  outside  diameter  of  the  gear 

N 
SM  =  —  -  and  sin  7  =  cos  0. 

N  +  2  N 

Hence, 


2  P  cos  <£      2  P  sin  j8 

iv 

sin  /3  =  ^—  —  cos  0  .....     (8) 

As  j8  is  known  from  Equation  (7),  N  can  be  obtained  by  solving 
(8).  The  value  of  ^V  thus  found  is  the  smallest  number  of  teeth 
permissible  in  the  larger  gear  if  interference  is  to  be  entirely 
eliminated. 


NOISY   GEARING  117 

Charts  for  Finding  Number  of  Teeth  in  Large  Gear.  —  The 

foregoing  solution  appears  cumbersome,  but  in  applying  it  to 
practice  the  only  equations  used  are  (6),  (7)  and  (8),  which  are 
easy  to  solve.  The  curves  corresponding  to  the  equations  for 
14^-  and  2o-degree  angles  of  involute  are  given  in  Figs.  6  and  7. 
One  curve  in  each  chart  was  arrived  at  by  solving  Equation  (8) 
and  was  plotted  with  values  of  sin  (3  as  abscissas  and  correspond- 
ing values  of  N  as  ordinates.  From  Equation  (6)  values  of  cot  a. 

N 
were  then  found  corresponding  to  given  values  of  —  (the  ratio). 

Knowing  a  and  $,  values  of  ft  were  then  obtained  from  Equation 

N 

(7)   corresponding  to  the  given  values  of  — .     Corresponding 

ifi 

values  of  sin  ft  were  then  found  from  a  table  of  sines,  and  the 

N 

second  curve  was  plotted  with  —  as  ordinates,  and  the  values  of 

w/ 

sin  ft  as  abscissas.  The  dotted  lines  on  each  chart  indicate  the 
course  to  be  traced  in  using  the  diagram.  The  most  usual  prob- 
lem will  be,  given  a  ratio,  to  find  the  most  suitable  number  of 
teeth  for  the  larger  gear.  In  solving  this  problem,  first  find  on 
the  left-hand  side  of  the  chart  a  figure  denoting  the  given  ratio, 
and  trace  horizontally  to  meet  curve  marked  "  Curve  I."  From 
this  point  trace  vertically  to  meet  "Curve  II,"  and  then  again 
horizontally  to  the  right-hand  side  of  the  chart,  where  the  correct 
number  of  the  teeth  for  the  larger  gear  will  be  found.  Thus  in 
the  example  chosen  it  was  required  to  find  the  correct  number 
of  teeth  in  the  larger  gear  of  a  pair  having  a  ratio  of  5  to  2.  This 
ratio  is  first  expressed  in  terms  of  unity,  viz.,  i\  to  i,  and  i\  is 
found  on  the  left-hand  side  of  the  chart.  The  dotted  line  is  then 
followed  horizontally  to  Curve  I,  and  then  vertically  to  Curve  II, 
and  finally  horizontally  to  the  right  side  of  the  chart.  In  Fig.  6, 
for  i4^-degree  involute,  the  number  of  teeth  is  found  to  be  65, 
and  in  Fig.  7,  for  2o-degree  involute,  38.  The  pinions  then  will 
have  26  and  15.4  teeth,  respectively.  Since  fractional  teeth  are 
impossible,  these  values  for  the  pinions  become  26  and  16,  and, 
therefore,  the  corresponding  gear  teeth  values  are  65  and  40. 

For  the  moment,  since  14!  degrees  has  been  practically  uni- 
versally adopted  as  the  standard  angle  of  involute,  our  attention 


n8 


SPUR   GEARING 


will  be  confined  to  Fig.  6.    Two  things  are  particularly  to  be 
noticed: 

i.  — As  the  ratio  increases,  the  number  of  teeth  in  the  pinion 
must  also  increase.    Thus  with  a  ratio  of  i  to  i  (equal  gears) 


•20 


0 

Machinery 


Fig.  6.    Chart  for  finding  Smallest  Number  of  Teeth  in  Large  Gear  for 
a  given  Ratio,  with  i^-degree  Pressure  Angle 

both  gears  should  have  22  teeth;  with  a  ratio  of  4  to  i  the  pinion 
must  have  120  -f-  4  =  30  teeth;  and  with  a  ratio  of  8  to  i  the 
value  must  be  increased  to  256-?- 8  =  32  teeth. 

2.  — The  number  of  teeth  in  common'use  for  pinions  is,  as  a 
general  rule,  far  too  small,  especially  for  the  high  ratios. 


NOISY  GEARING  119 

Possible  Methods  for  Avoiding  Interference.  —  The  signifi- 
cance of  these  two  points  is  very  great.  A  reference  to  recent 
specifications  for  gearing  will  show  that  to  a  certain  extent  this 
principle  has  been  grasped,  but  has  not  been  carried  sufficiently 
far.  Probably  the  greatest  field  at  the  present  day  for  gearing 
is  in  the  transmission  of  power  from  electric  motors,  and  it  is  here 
that  the  tendency  to  increase  the  number  of  teeth  in  the  gears  is 
most  evident.  Large  gears  with  teeth  of  fine  pitch  and  wide 
faces  are  used,  but  it  is  doubtful  whether  the  slight  advance 
which  has  been  made  in  this  direction  has  had  any  very  ap- 
preciable effect  in  reducing  the  noise  from  the  gearing.  There 
are  two  reasons  why  the  difficulty  cannot  be  wholly  dealt 
with  by  adopting  finer  pitches  in  order  to  increase  the  number 
of  teeth: 

1 .  —  From  consideration  of  strength,  if  fine  pitches  are  used, 
the  gear  face  must  be  correspondingly  increased.     Until  quite 
recently  face  widths  used  to  be  from  i\  to  3  times  the  circular 
pitch,  but  now  it  is  not  at  all  uncommon  to  find  5  or  6  times  the 
circular  pitch  used.     With  such  relatively  wide  faces  extreme 
accuracy  in  erection  is  necessary  in  order  to  secure  an  equal 
bearing  all  along  the  tooth  face.     Equal  accuracy  in  the  cutting 
is  also  most  important.     If  this  accuracy  does  not  obtain,  the 
whole  load  is  thrown  on  the  corner  of  a  tooth,  and  since  the  pitch 
is  small,  breakage  is  extremely  likely  to  occur.     Wear  in  the 
bearings  will  have  the  effect  described  under  cause  (4)  at  the 
beginning  of  this  chapter. 

2.  — High  ratios  are,  at  any  rate  with  electric  motors,  a  neces- 
sity, if  first  cost  of  installation  is  to  be  kept  down.     Now,  if  the 
correct  number  of  teeth  is  used  in  the  pinion,  the  number  of  teeth 
in  the  gear  becomes  so  great  that  difficulties  are  experienced  in 
the  cutting.     Even  on  the  hobbing  machine,  the  time  taken  by 
the  hob  in  traversing  the  circumference  of  the  gear  is  consider- 
able, and  on  this  account  there  is  strong  reason  to  believe  that 
local  heating  of  the  blank  is  introduced  with  consequent  errors 
of  pitch.     Such  errors  may  undoubtedly  be  reduced  to  a  mini- 
mum,   but   the   cost   of   production   is    thereby   considerably 
increased. 


120 


SPUR   GEARING 


These  two  objections  have  obviously  been  raised  by  the  prac- 
tical man,  who  alone  has  set  the  limit  for  the  number  of  teeth 
which  are  practically  advisable.  Fine  pitches  do  not  "look 
right,"  but  we  recognize,  of  course,  that  silence  is  important;  if 
it  is  only  to  be  obtained  by  the  adoption  of  finer  and  still  finer 
pitches,  the  question  of  practicability  arises.  Consequently,  if 
it  can  be  shown  that  we  may,  by  following  certain  simple  rules, 
return  once  more  to  the  coarser  pitches  and  secure  even  greater 


7.    Chart  for  finding  Smallest  Number  of  Teeth  in  Large  Gear  for 
a  given  Ratio,  with  2o-degree  Pressure  Angle 

degrees  of  silence  than  has  so  far  been  obtained  by  the  use  of  the 
finer  pitches,  a  great  deal  has  been  done  to  successfully  solve  this 
problem. 

It  is  evident  from  Fig.  7  that  one  way  of  reducing  the  number 
of  teeth  required  for  efficiency  is  to  increase  the  angle  of  the 
involute.  Thus  with  a  2o-degree  involute,  a  i6-tooth  pinion 
may  be  used  with  a  5  to  2  ratio,  as  compared  with  a  26-tooth 
pinion  with  a  i4j-degree  involute.  This  looks  promising,  and 


NOISY   GEARING 


121 


in  some  cases  has  been  adopted,  but  it  is  an  unfortunate  fact 
that  involutes  of  a  given  angle  of  obliquity  will  work  only  with 
others  having  the  same  angle;  consequently  2o-degree  involutes 
will  not  work  with  the  standard  i4|-degree  involutes.  It,  there- 
fore, will  be  seen  that  confusion  is  likely  to  result  from  varying 
the  angles,  and  in  any  case,  the  greater  advantage  of  interchange- 
ability  is  sacrificed.  Another  great  objection  to  the  adoption  of 


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Fig.  8.    Chart  for  finding  Smallest  Number  of  Teeth  in  Large  Gear  for 
Internal  Gearing,  with  14', -degree  Pressure  Angle 

increased  pressure  angles  is  that,  owing  to  the  greater  inclination 
of  the  path  of  contact,  the  pressure  tending  to  force  the  gear 
teeth  out  of  mesh  is  greatly  increased;  in  other  words,  gears  with 
teeth  of  high  obliquity  crowd  on  their  centers  to  a  considerably 
greater  degree  than  those  with  smaller  angles.  Many  designers, 
therefore,  believe  that  this  method  of  arriving  at  the  required 
result  is  not  advisable. 


122  SPUR   GEARING 

Proposed  Method  —  The  Shortened  Addendum.  —  A  method 
which  has  been  used  in  everyday  practice  with  marked  success, 
for  several  years,  may  be  explained  as  follows: 

Interference,  when  present,  is  due  to  the  addendum  of  the 
teeth  in  the  larger  gear  being  prolonged  beyond  the  limit  circle; 
therefore,  it  seems  logical  to  reduce  the  addendum  by  the  amount 
it  projects  beyond  this  circle.  It  is,  of  course,  necessary  to  see 
whether  by  so  doing  any  objectionable  features  are  introduced. 
A  point  which  appears  at  once  is  that  by  shortening  the  teeth, 
the  duration  of  contact  between  two  teeth  is  apparently  short- 
ened. If  the  duration  of  contact  is  decreased,  the  important 
question  to  consider  is  whether  the  shortened  duration  of  con- 
tact is  such  as  to  cause  one  tooth  to  come  out  of  mesh  before 
the  next  one  is  engaged.  Referring  to  Fig.  5  it  has  been  shown 
that 

N 

cot  a  =  — 7—          —  4-  tan  0.     .    • .     .     .     (6) 
n  sin  <j>  cos  0 

By  the  same  process  it  may  be  shown  that  if  the  angle  SOE 

is  called  6,  then  cot  6  =  -  — •  +  tan  0. 

N  sin  0  cos  0 

Now,  in  Fig.  5,  n  -r-  N  =  J,  and  0  is  14^  degrees.     Hence, 
cot  d  =  2.3206,  and  d  =  23  degrees  19  minutes. 

Angle  SOD  =  0  =  14  degrees  30  minutes.  Hence,  angle 
DOE  =  37  degrees  49  minutes. 

This  angle  may  be  taken  as  a  measure  of  the  angular  duration 
of  contact.  Around  the  circumference  of  the  pinion  there  are 
twelve  teeth  and  twelve  spaces.  It  therefore  follows  that 

37^2  __  x 

360  12 

where  x  equals  the  number  of  teeth  always  in  mesh.  In  the 
present  case  x  equals  1.26.  Hence,  it  will  be  seen  that  even 
in  the  extreme  case  chosen  one  tooth  will  not  come  out  of  mesh 
before  the  next  one  is  engaged. 

Life  of  Gears.  —  The  question  of  whether  the  life  of  the  teeth 
is  shortened  should  also  be  considered.  In  this  connection  it  is 


NOISY   GEARING  123 

necessary  to  point  out  that  the  action  between  teeth  of  true 
involute  profile  is  only  a  true  rolling  action  when  the  teeth  pass 
the  pitch  point  5,  Fig.  5.  At  any  other  points  in  contact  there  is 
more  or  less  slipping  between  them,  the  maximum  being  reached 
at  the  points  of  first  and  last  contact  and  the  minimum  at  the 
pitch  point.  When  the  teeth  first  make  contact  they  approach 
each  other  obliquely,  and  even  if  interference  is  absent  the  slip- 
ping is  very  great  and  causes  wear.  If  interference  is  present, 
the  effect  is  greatly  aggravated.  It,  therefore,  would  seem  that 
by  removing  the  point  of  the  tooth,  the  life  of  the  gear  is  actually 
prolonged  and  this  deduction  is  amply  borne  out  in  practice. 

When  it  becomes  necessary  to  use  a  smaller  number  of  teeth 
than  that  indicated  by  the  chart,  interference  may  be  avoided 
by  reducing  the  addendum  of  the  teeth  in  the  larger  gear  to 
what  it  would  be  if  the  number  of  teeth  found  from  the  chart 
were  used. 

For  instance,  suppose  that  the  number  of  teeth  given  by  the 
chart  is  180,  and  that  the  required  diametral  pitch  is  6.  The 
pitch  diameter  would  be  30  inches  and  the  addendum  £  inch. 
Suppose  also  that  120  teeth  is  the  largest  number  permissible. 
The  diameter  being  as  before  30  inches,  the  diametral  pitch 
would  have  to  be  4,  the  standard  addendum  being  J  inch.  With 
this  gear,  interference  would  occur,  but  if  according  to  the  rule 
the  addendum  were  reduced  to  J  inch,  no  trouble  would  be  met 
with.  The  actual  reduction  in  diameter  is  given  by  the  formula: 


(N-T\ 

=  2  \P  X  Nl 


where  A  =  amount  by  which  the  over-all  diameter  of  the  gear 

is  reduced; 

N  =  number  of  teeth,  found  from  chart; 
T  =  number  of  teeth  actually  used; 
P  =  diametral  pitch  actually  used. 

In  the  above  example: 

A  —      /i8o  —  i2o\  _  i 
2  V  4X  180  /     6 

which  agrees  with  what  has  previously  been  said. 


124  SPUR   GEARING 

General  Requirements.  —  When  carrying  out  this  method  in 
practice  difficulties  may  be  anticipated  from  cutting  the  teeth  if 
the  gear  has  been  previously  reduced  in  diameter;  but  as  the 
reduction  in  diameter  is  a  known  quantity  it  can  easily  be 
allowed  for  when  calculating  the  depth  of  the  cut.  If  difficulties 
are  experienced,  the  gears  may  be  left  with  standard  outside 
diameters  until  after  the  teeth  are  cut,  when  they  may  be  re- 
duced the  required  amount  in  the  hobbing  machine  itself. 

When  giving  this  method  a  trial,  it  is  not  advisable  that 
experiments  be  made  on  gears  that  have  already  been  in  use, 
because  if  wear  has  taken  place  the  results  may  be  misleading. 
The  probability  is  that  a  great  improvement  will  be  noticed  in 
the  running.  In  cases  where  it  has  been  tried,  no  instance  has 
ever  been  found  where  the  noise  did  not  decrease,  except  if  wear 
had  taken  place  to  any  extent,  when  other  conditions  are  met 
with,  and  the  trial  may  prove  misleading.  To  get  a  fair  idea 
of  the  results  obtained  by  this  method,  a  pair  of  gears  with  a 
ratio  of  3  or  4  to  i  should  be  made,  the  pinion  having  12  or  14 
teeth,  and  with  all  dimensions  standard.  Another  pair  should 
then  be  made  precisely  similar,  except  that  the  diameter  of  the 
larger  gear  should  be  corrected  as  suggested;  the  two  pairs 
should  then  be  carefully  erected  and  run  side  by  side  and  the 
difference  noted. 

It  may  be  pointed  out  that  since  the  addendum  of  the  gear 
tooth  has  been  reduced,  the  flank  space  of  the  pinion  may  be 
to  a  corresponding  extent  filled  in,  thus  shortening  the  pinion 
tooth  and  thereby  reducing  the  under-cut  and  greatly  increas- 
ing the  strength  of  the  tooth.  This  can  only  be  done  by  using 
a  special  hob  or  cutter  with  shortened  teeth,  and  unless  a  large 
number  of  duplicate  gears  are  to  be  cut,  the  expense  is  not 
warranted. 

It  should  be  pointed  out  that  if  the  gear  ratio  is  one  to  one 
(equal  gears),  and  the  number  of  teeth  to  be  used  is  less  than 
the  number  found  from  the  chart,  it  will  be  necessary  to  shorten 
the  teeth  of  both  gears.  This  being  so,  we  are  led  to  suppose 
that  possibly  with  other  ratios  it  may  become  necessary  to 
reduce  the  height  of  the  pinion  teeth,  if  a  low  number  of  teeth 


NOISY   GEARING  125 

is  used.  Such  is  the  case  in  practice,  and  it  may  be  demonstrated 
by  a  construction  for  the  pinion  similar  to  that  given  in  Fig.  5 
for  the  gear.  This  correction  of  the  pinion  may  be  necessary 
with  ratios  from  i  to  i  up  to  3  to  2,  but  since  the  ratios  between 
these  limits  are  less  often  used,  it  has  not  been  thought  necessary 
to  give  specific  formulas. 

Application  to  Rack  and  Pinion.  —  The  principle  may  be 
applied  to  rack  systems  in  which  case  it  can  be  shown  that  to 
avoid  interference: 

n  —  2       „ 

—  tan2  0  =  i 

2 

in  which  n  =  the  minimum  number  of  teeth  in  the  pinion; 
4>  =  the  pressure  angle  in  degrees. 

Solving  this  equation  for  <j>  =  14^  degrees,  we  obtain  n  = 
31.86  or  32  teeth.     A  smaller  number  of  teeth  may  be  used,  pro- 
vided the  rack  teeth  are  shortened  in  a  manner  similar  to  that 
described  for  external  gears. 
If     x  =  the  amount  to  be  cut  off  the  rack  teeth; 
N  =  the  number  of  teeth  to  be  used  in  the  pinion; 
<ft  =  the  pressure  angle; 
P  =  the  diametral  pitch; 

then  it  may  be  shown  that 

_  2  -  N  sin2  <ft 
2P 

It  will  be  noticed  that  if  N  is  made  =  31.86,  and  <ft  =  14^ 
degrees,  the  numerator  of  the  fraction  vanishes,  whence  x  =  o, 
which  agrees  with  what  has  previously  been  said. 

Application  to  Worm  Gearing.  —  It  is  obvious  that  when  deal- 
ing with  the  rack  and  pinion  we  have  also  disposed  of  the  worm 
and  worm-wheel.  It  follows  that  if  interference  is  to  be  avoided, 
a  worm-wheel  should  never  have  less  than  32  teeth.  Incidentally 
it  may  be  suggested  that  this  probably  explains  why  worm-wheels 
with  a  small  number  of  teeth  frequently  run  hot  in  spite  of  the 
fact  that  they  are  only  lightly  loaded. 

Application  to  Internal  Gear.  —  One  other  case  remains  to  be 
dealt  with,  viz.,  the  internal  gear.  It  has  so  frequently  been 


126 


SPUR   GEARING 


shown  that  there  is  a  certain  minimum  allowable  difference 
between  the  numbers  of  teeth  in  the  pinion  and  the  mating 
internal  gear  that  it  is  only  necessary  to  mention  it  in  passing, 
and  to  say  that  the  interference  which  occurs  if  this  rule  is 
infringed  upon  is  entirely  separate  from  the  interference  dealt 
with  throughout  this  chapter. 

The  argument  already  presented  for  spur  gears  may  be  equally 
well  applied  to  internal  gears.  As  with  the  spur  gear,  a  similar 
line  of  argument  results  in  three  formulas  which  naturally  bear 
a  strong  resemblance  to  Formulas  (6),  (7)  and  (8). 


FELT  PACKING 


Fig.  9.    A  Method  for  Eliminating  the  Noise  in  Crane  Gearing 


COt  a  = 


sin/3  = 


sin  4>  cos  0 
90°  +  <j>  —  a 

N 


N  -  2 


COS0 


(9) 

(10) 


where 


E  —  the  ratio  of  the  train; 

0  =  the  pressure  angle; 

N  —  the  number  of  teeth  in  the  internal  gear  which 
gives  the  best  all  around  results  for  that  par- 
ticular ratio. 

The  chart  shown  in  Fig.  8  combines  these  formulas  and  puts 
them  into  a  convenient  form  for  use.  A  smaller  number  of  teeth 
than  the  number  found  from  the  chart  may  be  used,  and  inter- 
ference is  avoided  as  before,  if  the  internal  teeth  are  shortened. 
It  must  be  noted  that  shortening  the  teeth  in  this  case  has  the 
effect  of  increasing  the  internal  diameter  of  the  gear  blank,  the 

(N  -  T\ 


increase  in  bore  being  equal  to  2 


V   PN 


NOISY   GEARING 


127 


in  which    N  =  the  charted  number  of  teeth  in  the  gear; 
T  =  the  actual  number  of  teeth  used; 
P  =  the  actual  diametral  pitch. 

Application  to  Bevel  Gears.  —  In  conclusion  it  may  be  pointed 
out  that  a  similar  line  of  reasoning  may  be  applied  to  bevel  gears, 
which  in  extreme  cases  may  need  correction  for  interference.  It 
is  hardly  necessary  to  state  that  the  actual  numbers  of  teeth  in 
the  gears  must  not  be  used  in  applying  the  formulas,  but  that  for 
these  values  must  be  substituted  the  developed  numbers  of  teeth 
obtained  in  a  manner  well  known  to  all  designers. 


Machinery 


Fig.  10.    Another  Method  to  prevent  Noisy  Gearing 

Make-shift  Methods  for  Avoiding  Noisy  Gearing.  —  A  little 
kink  for  eliminating  the  noise  of  gearing  is  shown  in  Fig.  9. 
Trouble  was  experienced  from  the  excessive  noise  made  by  the 
gearing  of  a  crane  placed  in  such  a  position  that  the  noise  was 
highly  objectionable.  Several  methods  were  tried  to  eliminate 
it.  Grease  and  oils  of  all  kinds  were  used  with  but  temporary 
success.  Finally  the  following  method  was  tried:  The  annular 
space  between  the  hub  and  the  rim  was  packed  with  wood.  This 
wood  butted  tightly  up  against  felt  pads  as  shown  in  the 
engraving.  The  pieces  of  wood  were  secured  to  each  other  by 
ordinary  wood  screws,  care  being  taken  not  to  have  the  heads 
project.  Good  hardwood  should  be  used,  and  rubber  might  be 
used  to  advantage  instead  of  felt,  except  for  exposed  outdoor 


128  SPUR   GEARING 

work.    This  method  eliminated  the  objectionable  noise  from  the 
gearing. 

The  method  shown  in  Fig.  10  is  even  more  efficient,  but  it  is 
also  far  more  expensive.  In  using  this  method  gears  of  less  than 
18  inches  in  diameter  are  fitted  with  two  sheets  of  tin  which 
enclose  the  space  between  the  hub  and  the  rim  of  the  gear.  This 
space  is  then  filled  with  sawdust  or  with  No.  4  shot,  the  idea  being 
to  eliminate  vibration  by  this  means.  In  some  cases,  it  has  been 
found  advantageous  to  use  a  mixture  of  shot  and  sawdust.  The 
sheets  of  tin  are  fastened  to  the  rim  and  hub  with  a  number  of 
small  screws,  as  shown  in  the  illustration.  When  the  diameter  of 
the  gears  exceeds  18  inches,  wooden  rings  are  used  in  place  of  the 
tin,  the  method  of  attachment  being  similar  in  either  case.  A 
felt  packing  is  used  to  prevent  the  sawdust  from  leaking  out. 
This  arrangement  has  the  further  advantage  of  closing  the  space 
between  the  spokes  of  a  wheel,  thus  making  it  impossible  for  a 
workman  to  get  his  arms  or  tools  caught  by  the  rotating  wheel. 


CHAPTER   VII 
DESIGN    OF    SPUR   GEARING 

Small  and  Medium-sized  Gears.  —  Typical  designs  of  gears 
for  different  materials  and  uses  are  shown  in  Figs,  i  to  6.  The 
one  shown  in  Fig.  i  is  the  usual  type  for  a  steel,  cast-iron,  brass, 
bronze  or  fiber  pinion.  All  of  its  proportions  are  determined  by 
the  gear  calculations  and  the  diameter  of  the  shaft  on  which  it  is 
mounted,  so  there  is  little  to  be  said  about  the  design.  When 
made  from  steel,  it  is  generally  formed  from  bar  stock  in  the  lathe 
or  screw  machine;  for  other  metals,  cast  blanks  are  mostly  used. 
It  is  the  practice  of  some  firms,  notably  the  Brown  &  Sharpe  Mfg. 
Co.  of  Providence,  R.  I.,  to  round  the  corners  of  all  pinion  and 
gear  blanks,  large  and  small,  as  shown  in  Fig.  2.  This  practice 
has  the  advantage  of  making  a  gear  more  easy  to  handle,  and  less 
liable  to  injury  in  case  it  is  accidentally  dropped;  it  gives  it  a 
neater  appearance  as  well. 

In  Fig.  3  is  shown  a  design  used  for  gears  having  a  greater 
number  of  teeth  than  those  ordinarily  known  as  " pinions." 
The  weight  has  been  lightened  by  recessing  the  sides  to  form  the 
web  shown,  connecting  the  rim  and  the  hub.  Gears  of  this  shape 
are  rarely  cut  from  bar  stock  as  the  removal  of  the  metal  to  form 
the  web  is  too  wasteful.  The  usual  practice  for  this  design  is  to 
make  the  blank  from  a  casting  or  a  drop  forging. 

Design  of  Large  Gears.  —  As  the  number  of  teeth  for  the  gear 
becomes  still  larger,  the  increasing  weight  of  the  gear  may  be 
lightened  by  cutting  out  relieving  spaces  in  the  web,  or  by 
abandoning  the  web  entirely,  and  using  arms  for  supporting  the 
rim.  This  scheme,  shown  in  Fig.  4,  with  arms  of  oval  section, 
is  the  one  best  adapted  for  small  and  medium-sized  gear  blanks, 
and  is  often  used  on  the  largest  work  as  well.  It  is  the  hand- 
somest of  all  designs  of  gear  wheels,  when  it  is  in  harmony  with 
the  rest  of  the  machine  to  which  it  belongs.  It  requires  somewhat 

129 


130 


SPUR  GEARING 


more  metal  for  the  same  strength  than  do  the  two  designs  next 
shown.  It  is  very  easily  molded.  Suitable  dimensions  for  gears 
of  various  sizes  made  in  this  way  are  tabulated  below  the  illus- 
tration. 

For  the  largest  gears,  made  of  steel,  cast-iron  or  bronze  castings, 
arms  of  +  or  H-section  are  largely  used.  Dimensions  for  gears 
of  these  types  are  given  in  Figs.  5  and  6.  In  these  designs,  the 
metal  is  so  distributed  as  to  give  a  high  degree  of  rigidity  for  the 
weight.  These  two  forms,  particularly  that  in  Fig.  6,  are  more 


Fig.2 


Machinery 


Figs,  i,  2  and  3.    Different  Types  of  Spur  Gears 

difficult  to  mold  than  those  previously  shown.  The  latter  form 
is  better  for  gears  whose  faces  are  very  wide  in  proportion  to  their 
pitch  than  either  of  the  two  in  Figs.  4  and  5. 

The  tabular  dimensions  given  for  the  various  forms  of  gears  are 
to  be  considered  as  suggestive  rather  than  authoritative.  The 
tables  have  been  in  constant  use  for  some  years,  however,  and 
have  proved  to  be  very  satisfactory.  "Draft,"  for  removing  the 
patterns  from  the  sand  in  molding,  is  not  shown  in  any  of  the 
illustrations.  It  should  be  provided  liberally,  and  should  be 
added  to  the  dimensions  given,  rather  than  taken  off. 


DESIGN  OF  GEARS 
Dimensions  of  Spur  Gears 


Dimensions  of  Spur  Gears  with  Oval  Arms 

Til 


r* — e — *r 


Fig.  4 


P  =  diametral  pitch,  P'  =  circular  pitch 
A  =  1.57  -T-  P  =  0.5  P';  F  -  2.00  -J-  P  ="0.65  P'; 

B  =  6.28  -T-  P  =  2.0  P';  G  =  W  +  0.025  pitch  diameter; 

C  =  3.14  -v-  P  =  P';  H  =  044  X  bore; 

Z>  =  4.71  -4-  P  =  1.5  P';  5'  =  B  +  %  inch  per  foot; 

E  =  0.79  -r  P  =  0.25  P';  C'  =  C  +  %  inch  per  foot. 


Dimensions  of  Spur  Gears  with  Ribbed  Arms  or  Arms  of  H-section 


Fig.  6 


P  =  diametral  pitch,  P'  —  circular  pitch 

A  =  1.57  •*•  P  =  0.5  P';  G  =  W  -f  0.025  pitch  diameter; 

B  =  7.85  -s-  P  =  2.5  P';  E  =  0.44  X  bore; 

C  =  0.04  -5-  P  -  0.3  Pf;  B'  «=  B  +  %  inch  per  foot. 

F  =  2.20  -T-  P  =  0.7  P'; 


132  SPUR   GEARING 

The  Governing  Conditions  in  the  Design  of  Gearing.  —  The 
problem  of  gear  design  is  one  of  materials  and  dimensions.  The 
considerations  on  which  the  designer  bases  his  choice  of  materials 
and  dimensions  are  those  of  strength,  durability,  efficiency, 
smoothness  of  action,  noiselessness  and  cost.  The  gear  cannot 
attain  perfection  in  all  these  particulars,  as  some  of  them  are 
mutually  hostile ;  the  item  of  cost,  especially,  has  to  be  sacrificed 
to  make  a  gain  in  any  other  direction.  The  problem  of  design  is 
thus  one  of  compromise,  and  the  designer  has  only  his  judgment 
to  rely  on  in  determining  the  relative  importance  of  the  various 
considerations. 

It  is  possible,  however,  to  lay  down  a  few  simple  rules  along 
this  line.  The  prime  consideration  is  that  of  strength.  If  the 
teeth  of  the  gear  are  not  strong  enough  to  transmit  the  pressure 
they  are  calculated  on  to  bear,  the  gear  will  break,  and  the  other 
virtues  it  may  possess  in  the  way  of  cheapness,  noiselessness,  etc., 
will  be  of  no  avail.  As  has  already  been  stated  in  a  preceding 
chapter,  in  gearing  subjected  to  occasional  use  only,  the  dur- 
ability is  sufficient  for  all  practical  purposes  if  the  strength  is 
sufficient;  but  there  is  a  possibility  that  gearing  transmitting 
power  at  high  speed  may  wear  out  before  it  breaks.  Where 
gearing  is  used,  as  in  automatic  machinery,  primarily  to  obtain 
certain  desired  movements  in  the  mechanism,  without  requiring 
the  transmission  of  any  great  amount  of  power,  the  question  of 
efficiency  is  not  of  prime  importance. 

On  the  other  hand,  when  the  main  object  of  a  pair  of  gears  is 
to  receive  so  many  horsepower  from  one  shaft,  and  transmit  as 
nearly  as  possible  the  same  amount  to  another  shaft,  the  question 
of  efficiency  becomes  one  worthy  of  the  most  careful  considera- 
tion. As  to  smoothness  of  action,  if  the  requirements  for  effi- 
ciency and  silence  have  been  met  the  gears  will  run  smoothly, 
without  shock  or  vibration.  Noiselessness,  as  a  problem  in 
design,  is  largely  a  matter  of  the  selection  of  materials,  always 
supposing  that  the  teeth  of  the  gears  are  formed  to  correct  tooth 
curves.  The  matter  of  cost  is  in  part  an  engineering  problem, 
and  in  part  a  commercial  one.  It  is  an  engineering  problem  in 
so  far  as  it  bears  on  the  problem  of  obtaining  the  greatest  perfec- 


DESIGN  OF   GEARS  133 

tion  in  the  other  particulars  enumerated,  with  a  given  expenditure ; 
and  it  is  a  commercial  problem  when  it  comes  to  determining 
how  great  a  degree  of  refinement  it  is  advisable  to  ask  the 
purchaser  of  the  finished  machine  to  pay  for. 

A  Model  Spur  Gear  Drawing.  —  After  designing  a  pair  of 
spur  gears  with  all  the  care  that  theoretical  knowledge  and  prac- 
tical experience  can  suggest,  there  still  remains  the  important 
task  of  recording  the  results  thus  obtained  on  a  drawing,  in  such 
a  form  that  they  will  be  intelligible  to  an  intelligent  workman. 
This  drawing  should  plainly  set  forth  every  point  of  informa- 
tion needed  for  the  successful  completion  of  the  work.  In  aim- 
ing at  this  mark,  the  student  should  study  the  model  drawing 
shown  in  Fig.  7.  The  arrangement  of  this  drawing,  and  the 
amount  and  kind  of  information  shown  on  it,  are  based  on  the 
drawing-room  practice  of  the  Brown  &  Sharpe  Mfg.  Co.,  Provi- 
dence, R.  I.  Some  changes  and  additions,  however,  have  been 
made.  The  design  of  the  wheel  in  Fig.  7  is  the  same  as  that 
shown  in  Fig.  6.  As  stated,  this  design  is  not  so  easily  molded 
as  that  in  Fig.  5,  but  it  is  the  most  suitable  form  for  gears  of  a 
comparatively  wide  face.  No  pattern  dimensions  are  .given.  A 
drawing  for  machine  shop  use  should  not  be  confused  by  a  maze 
of  dimensions  which  are  not  used  by  the  machinist.  The  pattern- 
maker can  be  taken  care  of  by  a  separate  drawing,  or  by  a  special 
blueprint  with  his  dimensions  put  on  in  yellow  pencil. 

Dimensions  on  Gear  Drawings.  —  The  dimensions  given  are, 
perhaps,  figured  somewhat  closer  than  is  required  on  work  of  this 
size.  The  important  dimensions,  such  as  the  outside  diameter 
and  the  center  distance,  on  which  depend  the  proper  fitting  of 
the  teeth  of  the  gear,  are  a  little  too  large  to  be  measured  with 
the  vernier  caliper,  but  they  should  surely  be  accurate  within 
0.005  mcn  —  a  limit  easily  attainable  by  a  skillful  workman. 

Where  a  definite  amount  of  allowable  variation  from  the  exact 
size  can  be  determined  on,  it  is  customary  in  interchangeable 
manufacturing  to  give  the  dimensions  with  maximum  and  mini- 
mum limits.  In  work  of  the  kind  shown  in  Fig.  7,  however, 
where  the  machine  is  " built"  rather  than  " manufactured,"  it 
is  not  usual  to  give  limits.  The  diameter  of  the  hole  in  the  hub 


134 


SPUR   GEARING 


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DESIGN  OF   GEARS 


135 


of  the  gear  is  given  as  "6"  Std."  on  the  drawing.  This  means 
that  the  hole  is  to  be  bored  and  reamed  until  it  will  make  a  good 
push  fit  for  a  standard  plug  gage  of  the  size  given.  It  will  be 
noted  that  all  the  dimensions  needed  by  the  workman  who  turns 
the  blank  are  appended  to  the  figure,  while  .those  needed  by  the 
workman  who  cuts  the  teeth  are  given  in  tabular  form. 

The  side  view  of  the  gear  on  the  left  is  needed  only  for  showing 
the  number  and  dimensions  of  the  arms  to  the  patternmaker. 
For  pinions  and  webbed  gears  it  may  be  omitted.  It  is  not 


,                                                                1 

.,       t 

*  2.58-  >H  31.42-30  TEETH  >j 

~\  \  T    1  r      A    r\  ]  T 

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IF           IT 

11                       ,        1    i 

RACK                                                                              DATA  FOR  CUTTING 

1  WANTED      MACH.   STEEL                                        DIAMETRAL  PITCH                            3 

FIN.OI.  ALL  OVCB                                                     WHOLE  DEPTH  OF  TOOTH      0.7190" 

ADDENDUM                                       0.3333" 

THICKNESS  OF  TOOTH             0.5236" 

NO.  OF  CUTTER                                 1 

Machinery 

Fig.  8.    Example  of  Properly  Dimensioned  Rack  Drawing 

necessary  in  standard  gearing  to  show  the  shape  of  the  teeth,  so 
the  side  view  is  given  as  showing  the  blank  before  the  teeth  are 
cut.  The  pitch  and  bottom  circles  are  represented  by  broken 
and  dotted  circles,  respectively.  The  shape  and  kind  of  teeth 
(whether  involute  or  cycloidal)  is  taken  care  of  by  the  cutter 
called  for  —  specified  by  its  proper  number  if  it  is  involute,  and 
by  its  letter  if  it  is  cycloidal. 

Rack  Drawing.  —  In  Fig.  8  is  shown  a  model  drawing  of  a  rack, 
which  is  self-explanatory.  Here,  as  in  the  previous  case,  the 
blank  dimensions  are  shown  attached  to  the  figure  of  the  rack, 
while  the  cutting  dimensions  are  tabulated. 


CHAPTER  VIII 

PRINCIPLES    OF    METHODS   FOR   CUTTING    SPUR 
GEAR  TEETH 

THERE  is  no  form  of  machine  tool  which  has  called  for  more 
ingenuity  in  design  than  the  gear-cutting  machine.  The  methods 
by  which  gears  may  be  cut  are  so  numerous,  the  requirements 
are  so  varied,  the  possible  application  of  ingenious  geometrical 
principles  through  the  mechanism  used  are  so  nearly  limitless, 
that  a  wonderful  variety  in  design  and  construction  has  been 
evolved,  affording  a  field  of  study  which  is  unparalleled  in  its 
interest  to  the  machine  designer. 

The  earliest  form  of  gear-cutting  machinery  to  attain  anything 
like  its  present  state  of  development  was  the  automatic  spur- 
gear  machine  using  a  milling  cutter  to  shape  the  tooth.  Later 
came  a  period  in  which  various  forms  of  bevel  gear  cutting 
machinery  were  evolved,  the  demand  being  stimulated  by  the 
necessities  of  the  chainless  bicycle  business.  More  recently,  the 
requirements  of  the  automobile  have  resulted  in  another  period 
of  inventive  activity,  which  has  resulted  in  the  development  of 
new  machines  and  processes  for  gears  of  all  kinds,  though  the 
bulk  of  the  attention  has  been  given  to  the  spur  and  bevel 
forms. 

The  Classification  of  Gear-cutting  Machinery.  —  Gear-cutting 
machinery  may  be  classified,  first,  according  to  its  product- 
There  are  four  main  divisions  in  this  classification,  separating 
from  each  other  the  machines  designed  for  cutting  spur,  spiral, 
bevel  and  worm  gearing,  respectively.  The  cutting  of  internal 
gears  and  racks  is  analogous  to  the  cutting  of  spur  gears,  and  is 
included  with  it.  Twisted  or  herringbone  gears  having  parallel 
axes  are,  in  general,  cut  in  the  same  way  as  spiral  gears,  though, 
as  gears,  they  belong  to  a  different  class.  Some  machines  are  so 
designed  as  to  be  capable  of  cutting  more  than  one  form  of  gear, 

136 


METHODS  OF  CUTTING  TEETH 


137 


but  it  is  only  done  by  making  certain  adjustments  or  using 
certain  attachments  which,  for  the  time  being,  convert  them  into 
machines  of  other  types.  The  best  example  of  a  machine  which 
covers  all  the  divisions  of  this  classification  is  the  universal 
milling  machine,  which  may  be  arranged  to  cut  the  teeth  in  any 
one  of  the  four  forms  mentioned. 

The  second  classification  of  gear-cutting  machinery  depends 
on  the  principle  of  action  involved.     The  five  methods  we  will 


Machinery 


Fig.  i.    The  Formed  Tool  Principle  of  Action  as  Exemplified  by  a 
Shaper  Tool  and  a  Milling  Cutter 

consider  are  —  the  formed  tool,  templet,  odontographic,  describ- 
ing-generating, and  molding-generating  methods.  This  classi- 
fication relates  particularly  to  the  way  in  which  the  tool  is  held 
and  guided  with  reference  to  the  work,  to  produce  the  desired 
form  for  the  tooth  surfaces. 

The  third  method  of  classification  relates  to  the  nature  of  the 
operation.  The  four  operations  we  will  consider  are  —  forming 
the  tooth  by  impression,  by  planing  or  shaping,  by  milling,  and 
by  grinding  or  abrasion. 


138  SPUR   GEARING 

The  Formed  Tool  Principle.  —  This,  the  simplest  and  most 
obvious  way  of  forming  a  gear  tooth,  is  illustrated  in  Fig.  i. 
The  gear  to  be  cut  is  held  firmly  on  a  work  arbor  which,  in 
turn,  is  firmly  supported  in  the  machine  in  such  a  way  that  it 
can  be  indexed  (or  rotated  through  an  angular  distance  corre- 
sponding to  one  tooth)  from  time  to  lime  as  occasion  requires. 
In  the  upper  part  of  the  cut  is  shown  a  planer  or  shaper  tool- 
post,  carrying  a  formed  tool  having  outlines  accurately  corre- 
sponding to  the  shape  of  a  space  between  two  of  the  teeth  it 
is  desired  to  form.  It  is  evident  that  this  formed  tool,  when 
mounted  in  the  toolpost  of  the  planer  or  shaper,  may  be  fed 
down  into  the  work  to  the  proper  depth,  in  which  case,  being  set 
centrally,  it  will  reproduce  its  outline  in  the  work.  The  work 
may  then  be  indexed,  and  the  operation  repeated  to  form  another 
tooth  space.  With  the  work  indexed  in  the  direction  shown  in 
the  cut,  four  tooth  spaces,  or  three  complete  teeth  have  been 
formed.  A  formed  milling  cutter  may  be  used  instead  of  the 
planer  or  shaper  tool.  This  is  shown  at  work  on  the  under  side 
of  the  blank.  It  reproduces  its  outline  in  the  work  in  the  same 
way  as  does  the  planer  tool,  being  rotated  in  the  direction 
indicated,  and  fed  through  the  work  at  the  same  time. 

The  Templet  Principle.  —  This  method  of  cutting  gears  is 
shown  in  Fig.  2.  As  in  the  previous  case,  the  work  is  held  on 
the  table  of  the  shaper.  A  templet  holder  is  also  mounted  on  the 
shaper  table,  carrying  a  templet,  having  a  surface  formed  to 
the  exact  outline  desired  for  the  finished  tooth.  The  tool-block 
is  disconnected  from  the  feed-screw  and  weighted  so  that  it  falls 
of  its  own  accord.  To  its  side  is  clamped  the  guide  shown, 
which  bears  on  the  templet.  As  the  table  of  the  shaper  is  fed 
to  the  right,  it  will  be  seen  that  the  curved  surface  of  the  templet 
will  raise  the  guide,  the  tool-block  and  the  tool,  in  such  a  fashion 
that  the  desired  outline  will  be  reproduced  on  the  gear  tooth. 
The  upper  surfaces  of  teeth  a,  b,  c  and  d  have  been  formed  in 
turn  in  this  way,  the  work  being  indexed  for  this  purpose  as  in 
the  previous  case.  With  the  primitive  arrangement  shown,  it 
will  be  necessary  to  reverse  the  work  on  the  arbor  to  form  the 
other  side  of  the  teeth.  Teeth  d  and  e  had  their  faces  finished  in 


METHODS  OF  CUTTING  TEETH 


139 


this  way,  tooth  d  being  thus  completely  formed.  It  will  be  seen 
that  obtaining  accurate  teeth  by  this  method  requires  —  first, 
an  accurate  templet;  second,  accurate  setting  of  the  templet  and 
tool  in  proper  relation  to  each  other;  and  third,  a  bearing  surface 
on  the  guide  of  exactly  the  same  shape  as  the  cutting  edge  of  the 
tool.  As  shown,  the  gear  to  be  cut  has  had  the  tooth  spaces 
roughed  out  to  shape,  so  that  the  finishing  operation  removes 
a  comparatively  small  amount  of  metal. 

The    Odontographic   Principle.  —  In   shaping   teeth   by   the 
odontographic  principle,  the  tool -is  guided  in  some  way  by  suit- 


DIRECTION  OF  FEED  O 


^ 

F  SHARER  TABLE  -  >  -..      .  . 

Machinery 


Fig.  2.    The  Templet  Principle  as  Arranged  to  be  Applied  to  a  Shaper 

able  mechanism,  to  closely  approximate  the  desired  tooth  out- 
line by  means  of  circular  arcs,  or  other  easily  obtained  curves. 
A  simple  example  is  shown  in  Fig.  3.  The  gear  to  be  cut  is  held 
and  indexed  as  in  the  two  previous  cases.  The  blank  has  had 
the  teeth  roughed  out  as  in  the  previous  case.  The  gear  to  be 
cut  has  involute  teeth.  With  teeth  of  this  form,  in  most  cases 
a  circular  arc  may  be  found  which  will  more  or  less  closely  ap- 
proximate the  true  outline.  Such  a  circular  arc  is  shown  at  xy, 
with  its  center  at  o.  The  radius  tool-holder  shown  has  its  center 
at  o  to  agree  with  that  of  arc  xy.  The  cutting  point  of  the  tool 
used  is  located  on  arc  xy.  It  will  be  seen  from  this,  that  when 


140 


SPUR   GEARING 


the  radius  tool  is  fed  from  position  T\  to  T^  by  the  feed  worm,  its 
point  will  follow  the  desired  arc  and  cut  the  desired  outline  for 
the  tooth.  By  this  means,  the  upper  surface  of  tooth  a  is  formed. 
The  same  surfaces  of  teeth  b,  c  and  d  have  previously  been  cut, 
as  well  as  the  opposite  faces  of  d  and  e,  tooth  d  being  completed. 
To  cut  the  opposite  faces,  the  work  may  be  reversed  on  the  arbor. 
The  Describing-generating  Principle.  —  This  principle  is 
shown  in  Fig.  4,  applied  to  the  shaping  of  involute  teeth.  The 


SHAPER  TABLE 


Machinery 


Fig.  3.    The  Odontographic  Principle  which  Approximately  Outlines 
the  Tooth  Form  by  Mechanical  Means 

cutting  of  involute  teeth  only  has  been  hitherto  shown  in  these 
examples,  owing  to  the  fact  that  in  other  cases,  as  in  this,  it  lends 
itself  most  readily  to  the  purposes  of  illustration.  The  involute, 
as  is  well  known,  is  the  curve  formed  by  a  point  in  a  cord  which 
is  being  unwrapped  from  the  periphery  of  a  circle.  In  the  cut, 
the  dotted  line  xy  shows  an  involute  generated  in  this  fashion 
from  the  base  circle  shown.  This  base  circle  is  formed  by  the 
periphery  of  the  rolling  disk,  which  is  firmly  connected  with  the 
gear  to  be  cut  through  the  work  arbor  on  which  both  are  mounted. 


METHODS  OF  CUTTING  TEETH 


141 


Unlike  the  previous  cases  considered,  the  work  arbor  in  this  case 
is  free  to  revolve  on  centers  without  being  restrained  by  an  in- 
dexing mechanism;  as  in  previous  cases,  the  blank  has  had  the 
teeth  roughed  out.  The  tool  used  is  a  shaper,  as  before.  To 
some  fixed  part  of  the  machine  is  clamped  the  tape  holder  shown. 
This  has  fastened  to  it  two  thin  flexible  metallic  tapes,  MI  and 
JI/2,  the  former  stretched  between  screw  Si  on  the  tape  holder 


/////////////^^^^^ 

DIRECTION  OF  FEED  OF  SHAPER  TABLE  > 


Machinery 


Fig.  4.    The  Describing-generating  Process  by  which  the  Point  of  the 
Tool  is  Constrained  to  Follow  the  Desired  Outline 

and  the  corresponding  screw  on  the  rolling  disk,  while  the  latter 
is  similarly  stretched  between  screws  S2  and  6*2.  By  this  means, 
it  will  be  seen  that  when  the  shaper  table  is  fed  in  the  direction 
indicated,  the  unwinding  of  MI  and  the  winding  of  M2  will  posi- 
tively roll  the  disk  and  the  work  with  it.  If,  now  a  tool  be  placed 
in  the  tool-block  of  the  shaper,  having  a  cutting  point  set  at  the 
same  height  as  the  middle  thickness  of  the  steel  tapes,  and  if  the 
table  be  fed  as  shown,  the  mechanism  will  constrain  the  tool 
point  to  cut  an  involute  on  the  side  of  the  tooth  of  the  gear  blank. 


142 


SPUR   GEARING 


When  the  tooth  is  at  c,  the  tool  will  be  at  Tc;  when  the  tooth  is  at 
b,  the  tool,  at  Tb,  will  have  cut  down  about  half  the  length  of  the 
face,  as  shown;  when  the  tooth  is  at  a,  its  outline  will  have  been 
completed  on  that  side  by  the  tool,  at  Ta.  The  way  in  which  the 
involute  is  generated  will  be  easily  understood,  when  it  is  seen 
that  the  cutting  point  of  the  tool  always  coincides  with  a  given 
point  y  in  tape  M%,  so  that  the  same  involute  as  is  generated  by 
this  point  in  the  unwinding  tape  is  reproduced  by  the  tool  point. 


Machinery 


5.     The   Molding-generating 
nciple  Applied   to  rolling  the 

Proper  Tooth  Form  in  a  Plastic 

Gear  Blank 


Fig.  6.  The  Same  Principle  em- 
ploying a  Cutter  having  a  Shap- 
ing Action  for  Cutting  Teeth  in  a 
Solid  Blank 


The  device  is  incomplete,  as  shown,  in  that  no  provision  is  made 
for  indexing.  In  this  case  the  gear  to  be  cut  and  the  rolling  disk 
have  to  be  indexed  with  relation  to  each  other,  so  as  to  present 
the  different  teeth  properly  for  the  tool  to  act  upon  them.  At 
d  is  shown  a  completed  tooth. 

The  Molding-generating  Principle.  —  This  method  of  making 
gears  depends  on  the  fact  that  in  a  set  of  interchangeable  gearing 
a  gear  formed  correctly  to  run  with  one  of  the  series  will  run  with 


METHODS  OF  CUTTING  TEETH  143 

all  of  the  series.  The  molding  process  consists  in  using  a  com- 
pleted gear  tooth  or  gear,  of  proper  shape,  to  form  the  others. 
Two  examples  of  this  are  shown  in  Figs.  5  and  6.  The  first  case 
supposes  a  forming  gear,  as  shown,  of  correct  shape.  The  blank 
to  be  formed  is  made  of  some  plastic  material  like  wax  or  clay. 
The  blank  and  the  forming  gear  are  mounted  on  arbors  at  the 
proper  distance  apart,  and  rotated  together  at  the  proper  speed 
ratio.  The  teeth  of  the  forming  gear,  pressing  into  the  plastic 
blank,  will  form  spaces  and  press  out  teeth  of  the  correct  shape 
to  mesh  with  itself>  or  with  any  other  gear  of  the  same  inter- 
changeable series. 

In  Fig.  6  the  blank  is  of  metal  or  other  non-plastic  material, 
and  the  forming  gear  is  replaced  with  a  forming  cutter,  having 
sharp  edges  of  exactly  the  same  outline.  The  blank,  which  in 
this  case  is  of  the  full  outside  diameter  of  the  gear  into  which 
it  is  to  be  made,  is  rotated  with  the  cutter,  as  in  Fig.  5.  The 
cutter  is  reciprocated  in  the  direction  of  its  axis  so  as  to  take  a 
series  of  cuts,  to  form  the  tooth  spaces  as  the  rotation  takes 
place.  The  principle  is  identical  with  that  shown  in  Fig.  5.  Of 
course,  the  cutter  has  to  be  fed  directly  in  to  the  proper  depth  to 
start  with,  before  the  rotating  commences. 

The  Four  Methods  of  Operation.  —  In  classifying  gear-cutting 
methods  by  the  operations  involved,  we  will  take  for  the  purpose 
of  illustration  the  molding-generating  method  as  applied  to  the 
spur  gear.  Later  on  we  will  see  how  the  same  operations  are 
applied  to  the  cutting  of  other  forms  of  gears,  by  other  methods. 
In  the  four  cases  shown  in  Figs.  7  to  10,  the  molding-generating 
is  done  by  a  rack  working  in  a  gear,  not  by  one  gear  working  in 
another,  as  in  Figs.  5  and  6. 

By  Impression.  —  Fig.  5  is  an  example  of  this  kind,  the  teeth 
in  the  plastic  blank  being  formed  by  the  impression  made  in 
it  by  the  forming  gear.  In  Fig.  7  the  same  thing  is  shown, 
except  that  the  forming  member  is  a  rack  which  has  shaped  the 
periphery  of  the  gear  with  which  it  meshes  into  correct  teeth,  as 
shown. 

By  Shaping  or  Planing.  —  In  Fig.  8  but  one  tooth  space  of 
the  gear  is  formed  at  a  time,  and  instead  of  using  a  rack  to  do 


144 


SPUR   GEARING 


the  forming,  a  tool  TI  may  be  used  having  an  outline  the  shape 
of  a  rack  tooth.  This  is  fed  along  horizontally,  and  the  gear  to 
be  cut  is  rotated  in  unison  with  it,  the  same  way  as  in  Fig.  7.  If 
tool  TI  is  given  a  cutting  movement  in  a  shaper,  the  spaces 
formed  will  be  of  exactly  the  right  shape  and  identical  with  those 


Fig.  1O. 
GRINDING  OR  ABRASION 

Machinery 


Figs.  7  to  10.    Pour  Methods  of  Operation  as  Applied  to  the  Molding- 
generating  Principle  of  Action 

formed  in  the  previous  case.  Each  of  the  spaces  will  have  to 
be  formed  in  the  same  way  one  after  another,  the  work  being  in- 
dexed with  reference  to  the  imaginary  rack,  to  bring  the  tool  into 
the  proper  position  for  each  of  them.  Instead  of  forming  both 
sides  of  a  space  at  one  operation,  as  with  tool  TI,  a  single  side 
tool  T2  may  be  used,  corresponding  with  one  side  only  of  the 


METHODS  OF  CUTTING  TEETH  145 

rack.  In  this  case  only  one  side  of  each  tooth  is  finished,  so  the 
tool  or  the  work  has  to  be  reversed,  after  which  the  other  sides 
are  completed. 

By  Milling.  —  Instead  of  using  a  planer  or  shaper  tool  to  match 
the  side  of  the  imaginary  rack  tooth,  a  milling  cutter  may  be  used, 
as  shown  in  Fig.  9.  In  this  case  the  gear  is  rotated,  and  the  mill- 
ing cutter  advanced  to  agree  with  the  advance  of  the  imaginary 
rack.  The  cutting  face  of  the  cutter  must  of  course  be  formed 
on  a  plane  surface,  as  shown.  This  arrangement  presents  some 
difficulties  when  the  gear  to  be  cut  has  a  wide  face,  since  the 
circular  cutter  will  cut  deeper  into  the  tooth  space  at  the  center 
than  it  will  toward  the  edges.  This  deepening  of  the  tooth  space 
at  the  center  does  not  affect  the  acting  tooth  surface,  and  so  is 
harmless  (except  possibly  in  the  case  of  the  generation  of  pinions 
having  a  small  number  of  teeth,  and  involute  outlines  of  low 
pressure  angle,  in  which  case  the  trouble  due  to  interference  is 
aggravated) .  The  larger  the  diameter  of  the  cutter,  as  compared 
with  the  face  of  the  gear,  the  less  is  the  trouble  on  this  score. 

By  Grinding  or  Abrasion.  —  In  Fig.  10,  the  milling  cutter  of 
Fig.  9  has  been  replaced  by  an  emery  wheel  of  similar  shape, 
having  a  plane  face  perpendicular  to  the  axis  of  the  wheel  spindle. 
The  action  on  the  work  is  identical  with  that  in  the  previous 
case,  subject  only  to  the  limitations  of  the  grinding  process,  such 
as  the  rapid  wearing  away  of  the  material  of  the  wheel,  involving 
the  necessity  for  constantly  truing  it  up.  Besides  this,  only  a 
small  amount  of  stock  can  be  removed  in  a  given  time,  as  com- 
pared with  the  rapidity  possible  with  a  milling  cutter.  The 
process  has  the  advantage  that  it  can  be  used  in  hardened  work. 

As  intimated,  each  of  these  various  operations  can  be  applied 
to  different  kinds  of  gears,  acting  according  to  different  principles, 
though  many  of  the  possible  combinations  are  impracticable. 

Machines  for  Forming  the  Teeth  of  Spur  Gears.  —  As  de- 
scribed in  the  previous  section,  spur  gear-teeth  may  be  formed 
in  any  one  of  five  ways  —  by  the  formed  tool  method,  the  tem- 
plet method,  the  odontographic  method,  the  describing-generat- 
ing method,  or  the  molding-generating  method.  The  extent  to 
which  these  various  schemes  have  been  applied  in  practical  use 


146  SPUR   GEARING 

varies  greatly.  The  formed  tool  method  is  at  once  the  most 
obvious  and  the  most  used  of  them  all.  The  templet  principle 
has  been  applied  to  a  limited  extent,  principally  for  gears  of  very 
large  size.  No  practical  application  of  the  odontographic  prin- 
ciple has  been  made  in  the  cutting  of  spur  gears.  The  only  ma- 
chine that  has  come  to  notice  involving  the  describing-generating 
process  was  one  invented  by  Mr.  Ambrose  Swasey,  and  in  use  a 
number  of  years  ago  in  the  shops  of  the  Pratt  &  Whitney  Co. 
This  was  not  used,  however,  for  making  gear  teeth,  but  for  making 
gear-tooth  cutters  —  before  the  days  of  the  formed  cutter,  which 
it  was  not  adapted  to  making.  The  molding-generating  process 
in  various  forms  has  received  a  wide  application,  second  only  to 
the  formed  tool  method. 

The  operations  available  for  the  formed  tool  method  are  — 
impression,  shaping  or  planing,  milling,  and  grinding  or  abrasion. 
Of  these,  the  impression  process  is  obviously  unsuited  for  practi- 
cal work.  The  shaping  or  planing,  and  the  milling  (particularly 
the  latter)  have  a  wide  range  of  application;  the  machines  used 
for  milling,  especially,  are  so  well  known  as  to  need  no  further 
comment.  In  the  case  of  the  operation  of  grinding  or  abrasion, 
only  a  single  machine  has  ever  been  built  embodying  the  formed 
tool  principle. 

Machines  using  Formed  Shaper  or  Planer  Tools.  —  The 
primitive  application  of  the  formed  tool  method  is  that  in  which 
a  gear  blank  is  mounted  on  index  centers  on  the  planer  or  shaper 
table,  and  has  its  teeth  cut  by  a  tool  having  an  outline  corre- 
sponding to  the  desired  tooth  space.  In  this  operation  the  tool 
is  fed  by  hand  to  the  proper  depth  and  withdrawn.  The  work 
is  then  indexed  for  a  second  cut,  the  tool  is  fed  down  again,  and 
the  operation  is  repeated,  until  the  gear  is  finished.  This  is 
shown  diagrammatically  in  the  upper  part  of  Fig.  i.  It  is  the 
simplest  method  of  cutting  a  gear  which  has  to  be  made  im- 
mediately, and  for  which  formed  milling  cutters  are  not  available. 
It  also  has  its  application  in  the  case  of  gears  of  unusual  size. 
Under  these  circumstances,  however,  the  machine  used  is  gener- 
ally a  slotter  instead  of  a  planer  or  shaper.  A  formed  tool  is 
fastened  in  the  toolpost  of  the  machine,  while  the  work  is  clamped 


METHODS   OF   CUTTING  TEETH 


147 


to  the  revolving  table.  The  indexing  is  done  by  such  means  as 
may  be  provided,  usually  a  worm  and  worm  gear  or  a  master 
wheel.  The  Gleason  and  Newton  templet  machines  also  may  be, 
and  doubtless  often  are,  used  in  the  same  way. 

The  Molding-generating  Milling  or  Robbing  Machine.  —  The 
most  widely  used  process  involving  the  milling  operation  of 


GEAR  BEING  CUT 


f  HELIX  ANGLE 
<      OF  HOB  AT 
PITCH  SURFACE 


.WORM  REPRESENTING  THE  HOB 
WHICH  IS  CUTTING  THE  GEAR 


ft  IMAGINARY  RACK  WHICH 
FORMS  THE  GEAR  BY  THE 
MOLDING  GENERATING  PRO- 
<  CESS -ITS  TEETH  COINCIDE 
I  WITH  THOSE  OF  THE  HOB 
J    WHEN    THE    LATTER    18 
SET   AS    SHOWN 


GEAR  BEING  CUT 


Machinery 


Fig.  iz.    Diagram  illustrating  the  Principle  of  the  Hobbing  Process  of 
Forming  Spur  Gears 

molding-generating  is  the  hobbing  process.  The  principle  of 
this  method  is  shown  diagrammatically  in  Fig.  n.  Here  we 
have  an  imaginary  rack  meshing  with  a  gear,  and  molding  its 
teeth  in  the  same  way  as  in  Figs.  7  to  10.  The  teeth  of  this  rack, 
shown  in  dotted  outline,  coincide  with  the  outlines  of  a  hob, 
shown  in  full  lines,  which  has  been  set  at  such  an  angle  as  to 
make  the  teeth  on  its  front  side  parallel  with  the  axis  of  the  gear. 


148  SPUR  GEARING 

In  other  words,  it  has  been  set  at  the  angle  of  its  helix,  measured 
at  the  pitch  line.  It  will  be  seen  that  the  teeth  of  the  hob,  when 
set  in  this  position,  correspond  with  the  teeth  of  the  rack.  If, 
now,  the  hob  and  blank  be  rotated  at  the  ratio  required  by  the 
number  of  threads  in  the  hob  and  the  number  of  teeth  in  the  gear, 
this  movement  will  cause  the  teeth  of  the  hob  to  travel  length- 
wise in  exactly  the  same  way  as  the  teeth  of  the  imaginary  rack 
would  travel,  if  in  mesh  with  the  gear  whose  teeth  are  to  be  cut. 
It  will  thus  be  seen  that  the  hob  fulfills  the  requirements  neces- 
sary for  molding  the  teeth  of  the  gear  to  the  proper  form.  In 
practice  the  hob  is  rotated  in  the  required  ratio  with  the  work, 
and  fed  gradually  through  it  from  one  side  of  the  face  to  the  other. 
When  it  has  passed  through  once,  the  work  is  completed. 

Of  the  great  number  of  machines  built  during  the  past  few 
years  involving  this  principle,  many  are  arranged  for  cutting 
spiral  gears  as  well  as  spur  gears.  Of  course,  all  of  the  machines 
capable  of  cutting  spiral  gears  are  capable  of  cutting  spur  gears 
also.  The  spiral  gear-hobbing  machine  bears  about  the  same 
relation  to  the  plain  spur  gear-hobbing  machine  that  the  uni- 
versal does  to  the  plain  milling  machine.  The  added  adjust- 
ments and  mechanism  required  in  each  case  tend  to  somewhat 
limit  the  capacity  of  the  machine  in  taking  heavy  cuts,  though 
they  add  to  its  usefulness  by  extending  the  range  of  work  it  is 
capable  of  performing. 

Requirements  of  Gear  Robbing  Machines.  —  The  require- 
ments of  the  successful  gear  hobbing  machine  are : 

First.     A  frame  and  mechanism  of  great  rigidity. 

Second.     Durable  and  powerful  driving  mechanism. 

Third.     Accurate  indexing  mechanism. 

The  first  requirement  is  one  of  great  importance,  not  only 
in  its  influence  on  the  heaviness  of  the  cut  to  be  taken  and  the 
consequent  output  of  work,  but  on  the  matter  of  accuracy  as 
well.  The  connection  between  the  hob  and  the  work,  through 
the  shafts  and  gearing,  is  liable  to  be  so  complicated  that  the 
irregular  cutting  action  of  the  hob  produces  torsional  deflections 
in  the  connecting  parts,  leading  to  serious  displacement  from 
the  desired  relation  between  the  hob  and  the  teeth  being  cut. 


METHODS  OF  CUTTING  TEETH  149 

This  displacement  from  the  desired  position  results  in  teeth  of 
inaccurate  shape,  weak  and  noisy  at  high  speeds. 

In  its  effect  on  the  output,  rigidity  is  even  more  important 
in  the  hobbing  machine  than  in  the  orthodox  automatic  gear 
cutter.  A  heavier  cut  is  taken,  since  a  greater  number  of  teeth 
are  cutting  on  the  work  at  once.  The  number  of  joints  between 
the  cutter  and  the  work-supporting  table  and  spindle  must, 
therefore,  be  reduced  to  a  minimum,  and  the  matter  of  overhang 
both  for  the  work  and  the  cutter  must  be  carefully  looked  out 
for.  The  reduction  of  overhang  is  hampered  at  the  cutter  head 
by  the  necessity  for  a  strong  drive  and  an  angular  adjustment. 
In  the  case  of  the  work-supporting  parts,  it  is  difficult  to  bring 
the  cutting  point  close  to  the  bearing  on  account  of  the  necessity 
for  plenty  of  clearance  below  the  work  for  the  hob  and  its  driving 
gear. 

The  matter  of  design  of  the  driving  mechanism  for  the  hob 
and  the  work  is  a  difficult  one.  Not  only  must  it  be  rigid  for 
the  sake  of  accuracy,  as  previously  explained,  but  careful  atten- 
tion must  be  given  to  durability  as  well.  It  requires  great  skill 
to  design  a  durable  mechanism  for  the  purpose  within  the 
limitations  imposed  —  in  the  cutter  head  by  the  necessity  for 
reducing  tbe  overhang,  and  in  the  work  table  by  the  high  speed 
required  for  cutting  small  gears. 

Since  the  indexing  wheel  works  constantly  and  under  con- 
siderable load,  both  the  wheel  and  worm  must  be  built  of  such 
materials  as  will  preserve  their  accuracy  after  long  continued 
use.  Particular  attention  should  be  given  to  the  homogeneity 
of  the  material  of  the  index  worm-wheel,  to  make  sure  that  it 
does  not  wear  faster  on  one  side  than  on  the  other. 

The  field  of  the  hobbing  process  for  cutting  spur  gears  has  not 
yet  been  definitely  determined.  In  some  work  it  appears  to 
have  certain  advantages  over  the  usual  type  of  automatic  gear- 
cutting  machine,  while  in  other  cases  it  falls  behind.  It  will 
doubtless  require  continued  use,  with  a  variety  of  work,  and  for 
a  considerable  length  of  time,  to  determine  just  what  cases  are 
best  suited  for  the  hobbing  machine,  and  what  for  the  machine 
with  the  rotating  disk  cutter.  It  is  not  probable  that  in  the 


SPUR   GEARING 


future  either  of  them  will  occupy  the  field  to  the  exclusion  of 
the  other. 

Designing  a  Hob  for  Robbing  Spur  Gears.  —  In  explaining 
the  methods  used  in  the  design  of  hobs  for  spur  gears,  it  is  best 
to  assume  a  practical  example.  Suppose  that  the  gears  are  to 
be  cast  iron,  with  120  teeth,  16  diametral  pitch  and  f  inch  width 
of  face.  The  pitch  diameter,  hence,  is  y|  inches.  The  hole 
in  the  hob  for  the  spindle  is  to  be  i|  inch  in  diameter  with 
a  J-inch  square  keyway,  the  hob  to  be  run  at  high  speed. 
Mr.  John  Edgar,  in  MACHINERY,  November,  1912,  gives  the 
following  solution  of  the  problem. 


< 0.-1963- H 


SlacMnery 


Fig.  12.    Standard  Hob  Tooth  Dimensions 

Form  and  Dimensions  of  Tooth.  —  The  first  thing  to  be  settled 
is  the  form  and  dimensions  of  the  tooth  or  thread  section  of  the 
hob.  If  the  form  is  to  be  the  standard  shape  for  the  involute 
system  with  a  14^-degree  pressure  angle,  the  dimensions  of  the 
hob  tooth  would  be  as  shown  in  Fig.  12.  A  modification  of  this 
shape  may  in  some  cases  be  advisable,  and  will  be  referred  to 
later  in  this  chapter.  The  standard  rack-tooth  shape  with 
straight  sides,  as  shown  in  Fig.  12,  however,  is  the  easiest  to 
produce,  and  it  is  entirely  satisfactory,  unless  gears  with  a  very 
small  number  of  teeth  are  to  be  cut. 

The  circular  pitch  corresponding  to  16  diametral  pitch  is 
0.1963  inch,  which  is  obtained  by  dividing  3.1416  by  16.  The 
thickness  of  the  tooth  on  the  pitch  line  is  one-half  of  the  cir- 
cular pitch,  or  0.0982.  The  height  of  the  tooth  above  the 


METHODS   OF   CUTTING  TEETH  151 

pitch  line  is  equal  to  the  reciprocal  of  the  diametral  pitch  plus 
the  clearance,  which  latter  is  equal  to  o.i  of  the  thickness  at  the 
pitch  line.  Hence,  the  height  of  the  tooth  above  the  pitch  line 
equals  0.0625  +  0.0098  =  0.0723  inch.  This  distance  equals 
the  space  in  the  gear  below  the  pitch  line. 

The  depth  of  the  tooth  of  the  hob  below  the  pitch  line  is 
usually  made  greater  than  the  distance  from  the  pitch  line  to 
the  top  of  the  tooth.  The  extra  depth  should  be  equal  to 
from  one-half  to  one  times  the  clearance.  On  small  pitches, 
one  times  the  clearance  is  not  too  great  an  allowance,  and, 
therefore,  the  depth  below  the  pitch  line  is  made  equal  to 
0.0723  +  0.0098  =  0.0821,  making  the  whole  depth  of  tooth 
equal  to  0.1544.  The  extra  depth  at  the  root  of  the  thread  is 
to  allow  for  a  larger  radius  at  the  root,  so  as  to  prevent  cracking 
in  hardening.  The  radius  may  then  be  made  equal  to  two 
times  the  clearance,  if  desired.  In  the  illustration,  however, 
the  radius  is  made  equal  to  o.i  of  the  whole  depth  of  the  tooth. 
The  top  corner  of  the  tooth  is  rounded  off  with  a  corner  tool  to 
a  radius  about  equal  to  the  clearance,  or  say  o.oio  inch.  This 
corner  is  rounded  to  avoid  unsightly  steps  in  the  gear  tooth 
flank  near  the  root.  Having  obtained  the  hob  tooth  dimensions, 
the  principal  dimensions  of  the  hob  may  be  worked  out  with 
relation  to  the  relief,  the  diameter  of  the  hole  and  the  size  of 
the  keyway. 

Relief  of  Hob  Tooth.  —  The  proper  relief  for  the  tooth  is  a 
matter  generally  decided  by  experience.  We  may  say  that, 
in  general,  it  should  be  great  enough  to  give  plenty  of  clearance 
on  the  side  of  the  tooth,  and  on  hobs  of  i4|-degree  pressure 
angle  the  peripheral  relief  is,  roughly  speaking,  about  four  times 
that  on  the  side.  For  cutting  cast  iron  with  a  hob  of  the  pitch 
in  question,  a  peripheral  relief  of  o.  1 20  inch  will  give  satisfactory 
results;  for  steel,  this  clearance  should  be  somewhat  increased. 
The  amount  of  relief  depends,  necessarily,  also  upon  the  diam- 
eter of  the  hob. 

With  a  peripheral  relief  of  0.120  inch,  the  greatest  depth  of 
the  tooth  space  in  the  hob  must  be  0.1544 .+  0.120  =  0.2744. 
The  gash  will  be  made  with  a  cutter  or  tool  with  a  2o-degree 


152 


SPUR   GEARING 


included  angle,  •/$  inch  thick  at  the  point,  and  so  formed  as 
to  produce  a  gash  with  a  half-circular  section  at  the  bottom. 
The  depth  of  the  gash  should  be  T\  inch  deeper  than  the  greatest 
depth  of  the  tooth  space,  or  about  ^  inch. 

Thickness  of  Metal  at  Keyway.  —  The  radius  of  the  hob  blank 
should  be  equal  to  f  +  J  +  $  +  the  thickness  of  the  stock 
between  the  keyway  and  the  bottom  of  the  flute.  If  we  use  a 
3 -inch  bar  we  can  turn  a  hob  blank  2\  inches  in  diameter  from 


REFACE  EACH  TOOTH 
AFTER  BACKING -OFF 
TO  REMOVE  EFFECTS 

OF  SPRING  IN  |TOOL 


Machinery 


Fig.  13.    Hob  with  Twelve  Gashes  or  Flutes 

this,  which  would  allow  sufficient  stock  to  be  turned  from  the 
outer  portion  of  the  bar  to  remove  the  decarbonized  surface. 
If  we  make  the  blank  2|  inches  in  diameter  we  have  -^  inch  of 
stock  over  the  keyway,  which  is  sufficient. 

Number  of  Flutes.  —  The  number  of  gashes  or  flutes  depends 
on  many  factors.  In  Fig.  13  is  shown  an  end  view  of  a  hob  with 
twelve  gashes.  This  number  gives  plenty  of  cutting  teeth  to 
form  a  smooth  tooth  surface  on  the  gear  without  showing  promi- 
nent tooth  marks.  A  larger  number  of  gashes  will  not,  in  prac- 


METHODS  OF  CUTTING  TEETH 


153 


A- 


tice,  give  a  better  tooth  form,  but  simply  increases  the  liability 
to  inaccuracies  due  to  the  forming  process  and  to  distortion  in 
hardening.  This  number  of  gashes  also  leaves  plenty  of  stock 
in  the  teeth,  thus  insuring  a  long  life  to  the  hob. 

Straight  or  Spiral  Flutes.  —  The  question  whether  the  gashes 
should  be  parallel  with  the  axis  or  normal  to  the  thread  helix 
is  one  that  is  not  easily  answered.  It  must  be  admitted  that 
when  the  angle  of  the  thread  is  great,  the  cutting  action  at  both 
sides  of  the  tooth  is  not  equal 
in  a  hob  with  a  straight  gash; 
but  in  cases  of  hobs  for  fine 
pitch  gears,  where  the  hobs 
are  of  comparatively  large 
diameter,  thus  producing  a 
small  thread  angle,  the  paral- 
lel gash  is  more  practical  be- 
cause it  is  much  easier  to 
sharpen  the  hobs,  and  the 
long  lead  necessary  for  spiral 
gashes,  in  such  cases,  is  not 
easily  obtained  with  the  reg- 
ular milling  machine  equip- 
ment. However,  when  it  is 
desired  to  obtain  the  very  best 
results  from  nobbing,  especi- 
ally in  cutting  steel,  the  gash 
should  be  spiral  in  all  cases 

when  the  thread  angle  is  over  2\  or  3  degrees.  In  our  case 
the  thread  angle  figured  at  the  pitch  diameter  of  the  blank  is 
equal  to  i  degree  22  minutes;  hence,  straight  flutes  are  not 
objectionable. 

Threading  the  Hob.  —  The  linear  pitch  of  the  hob  and  the 
circular  pitch  of  the  gear,  when  considered  in  action,  are  to  each 
other  as  i  is  to  the  cosine  of  the  thread  angle.  In  the  present 
case  they  do  not  differ  appreciably  and  may  be  considered  as 
equal.  In  cases  where  the  difference  is  over  0.0005,  the 
linear  pitch  should  be  used. 


Machinery 


Fig.  14.    Threading  Tool  for  Hob 


154 


SPUR   GEARING 


The  change-gears  for  the  lathe  may  be  figured  by  the  formula : 

Gear  on  lead-screw    _  lead  of  lead-screw 
Gear  on  stud  linear  pitch  of  hob 

On  a  lathe  with  a  lead-screw  of  six  threads  per  inch,  or  a  lead 
of  £  or  0.1667  inch,  the  gears  that  would  give  accurate  enough 
results  for  the  present  hob  would  be  28  teeth  on  the  lead-screw, 
and  33  teeth  on  the  stud. 

Thread  Relieving  Tool.  —  In  Fig.  14  is  shown  the  hob  thread 
relieving  tool.  The  front  of  the  tool  is  relieved  with  a  20-degree 
rake  for  clearance.  The  sides  are  ground  straight  at  a  14^- 
degree  angle  to  form  the  sides  of  the  thread,  and  are  at  an  angle 


r 


Fig.  15.    Special  Hob  Tooth  Dimensions 

of  i  degree  22  minutes  with  the  vertical  to  clear  the  sides  of  the 
thread.  A  tool  made  like  this  can  be  sharpened  by  grinding 
across  the  top  without  losing  its  size  or  form.  If  the  gashes  were 
made  on  the  spiral,  the  top  of  the  tool  should  be  ground  to  the 
angle  of  the  thread,  as  shown  by  the  dotted  line  AB.  In  cases 
where  the  angle  of  the  thread  is  considerable,  the  angle  of  the 
sides  of  the  tool  must  be  corrected  to  give  the  proper  shape  to 
the  hob  tooth.  (See  MACHINERY,  May,  1905,  or  MACHINERY'S 
Reference  Book  No.  32,  " Formula  for  Planing  Thread  Tools.") 
The  point  of  the  tool  should  be  stoned  to  give  the  proper  radius 
to  the  fillet  in  the  bottom  of  the  hob  tooth  space. 

Heat-treatment  of  Hob.  —  The  best  practice  in  making  the 
hob  is  to  anneal  it  after  it  has  been  bored,  turned,  gashed  and 
threaded,  the  annealing  taking  place  before  relieving  the  teeth. 


METHODS   OF   CUTTING  TEETH  155 

Before  hardening,  the  hob  ought  to  be  re-gashed  or  milled  in  the 
groove,  removing  about  3^  inch  of  stock  from  the  front  side  of 
the  tooth  to  eliminate  chatter  marks  and  the  effect  due  to  the 
spring  in  the  tool,  which  always  leaves  the  front  edge  of  the  teeth 
without  relief.  In  hardening,  do  not  attempt  to  get  the  hob  too 
hard,  as  the  required  high  heat  and  quick  cooling  would  distort 
the  teeth  badly. 

Modified  Tooth  Shape  in  Hob.  —  In  case  the  i2o-tooth  gear 
is  to  run  with  a  pinion  of  a  small  number  of  teeth  and  is  the 
driver,  as  in  small  hand  grinders  where  gears  of  this  size  are 
often  used,  it  would  be  advisable  to  make  the  tooth  shape  as 
shown  in  Fig.  15.  This  shape  will  obviate  undercutting  in  the 
pinion  and  relieve  the  points  of  the  teeth  in  the  gear  so  as  to 
obtain  a  free-running  combination.  This  shape  is  more  difficult 
to  produce  and  requires  more  care  in  forming.  If  the  hob  is 
made  of  high-speed  steel,  it  should  run  at  about  115  revolutions 
per  minute  for  cutting  an  ordinary  grade  of  cast  iron  with  a  feed 
of  Yg  inch  per  revolution  of  the  blank.  The  feed  may  be  in- 
creased considerably  if  the  gear  blank  is  well  supported  at  the 
rim.  The  best  combination  of  speeds  and  feeds  in  each  case  can 
be  found  only  after  considerable  experimenting. 

Interchangeability  of  Hobbed  and  Milled  Gears.  —  There  is 
always  an  objection  to  changing  existing  methods  in  shop  practice 
when  the  change  necessitates  discarding  established  standards 
and  valuable  tools  and  fixtures.  Whether  such  a  change  will  be 
profitable  or  not  is  a  question  that  requires  a  close  study  of  the 
conditions  in  each  case.  When  the  change  means  an  improve- 
ment in  the  quality  of  the  product,  the  cost  of  the  tools  and 
fixtures  should,  of  course,  be  a  secondary  consideration.  When 
the  question  is  mainly  one  of  quantity,  the  problem  must  be 
solved  on  a  cost  basis  only. 

Another  factor  to  be  considered,  however,  is  that  of  inter- 
changeability.  This  is  a  most  important  item  in  the  case  of  a 
product  in  connection  with  which  renewals  are  constantly  being 
made.  Many  improvements  in  design  and  in  methods  of  manu- 
facture are  sacrificed  in  deference  to  the  demands  for  interchange- 
ability.  In  the  case  of  gears,  interchangeability  is  supposed  to 


156  SPUR   GEARING 

be  rigidly  adhered  to,  but  while  we  have  a  standard  which  is 
supposed  to  produce  interchangeable  gears,  there  are  so  many 
variations  of  the  standard,  due  to  the  secret  forms  established  by 
different  manufacturers  of  cutters,  that  it  is  necessary  in  many 
cases  to  adhere  to  one  make  of  tools  if  interchangeability  is  to  be 
maintained  in  any  degree.  Many  manufacturers  have  installed 
the  hobbing  machine  in  the  desire  to  reduce  the  cost  of  gearing, 
only  to  encounter  the  non-interchangeability  of  the  product  of 
the  hobbing  machine  with  the  milled  tooth  gear;  this  has  been 
the  cause  of  turning  many  against  the  hobbing  machine,  through 
no  fault  of  the  process  itself. 

Variations  from  the  True  Involute  Tooth  Shape.  —  The  form 
of  the  standard  tooth,  as  adopted  by  the  cutter  manufacturers, 
is  not  the  true  involute,  but  an  improvised  form  built  around  the 
involute  as  a  basis.  The  deviation  from  the  involute  is  necessary 
for  several  reasons: 

1.  The  inability  of  the  formed  milling  cutter  to  mill  an  under- 
cut tooth. 

2.  The  necessary  alteration  in  the  form  of  the  point  of  the 
mating  tooth  caused  by  the  fullness  of  the  milled  tooth  below  the 
pitch  line. 

3.  The  desire  to  make  the  contact  of  the  approach  as  gradual 
as  possible  by  a  slight  easing  off  of  the  form  at  the  point  of  the 
tooth;  this  provides  against  the  slight  variation  in  the  form  of 
the  tooth  due  to  irregularities  in  the  division  of  the  space  and  to 
the  elasticity  of  the  material. 

4.  The  interference  in  gears  with  thirty-two  teeth  or  less  when 
in  mesh  with  those  of  a  greater  number  of  teeth.     As  the  14^- 
degree  formed  gear-cutters  are  based  on  the  twelve-tooth  pinion 
with  radial  flanks,  a  rack  tooth  to  mesh  with  this  radial  flank 
tooth  can  be  made  with  the  straight  sides  extending  only  to  a 
point  0.376  inch  outward  from  the  pitch  line  in  a  rack  of  one 
diametral  pitch.     The  remainder  of  the  tooth  must  be  eased  off 
from  this  point  outward,  sufficiently  to  clear  the  radial  flank  of 
the  pinion  tooth.     This  rounding  off  of  the  rack  tooth  may  be 
made  by  using  the  cycloidal  curve  from  the  interference  point, 
with  a  rolling  circle  of  a  diameter  equal  to  that  of  the  twelve- 


METHODS  OF   CUTTING  TEETH 


157 


tooth  pinion.  A  circular  arc  tangent  to  the  tooth  side,  drawn 
from  a  center  on  the  pitch  line  at  the  point  of  intersection  of  the 
normal  to  the  tooth  side  at  the  point  of  interference,  will  be  a 
near  approximation  to  the  cycloidal  curve. 

The  hobs  used  extensively  today  are  not  made  to  produce 
teeth  in  any  near  approximation  to  the  shape  produced  by  the 
milling  cutter.  The  only  correction  that  is  made,  in  many  cases, 
is  to  make  the  teeth  of  the  hob  a  trifle  fuller  at  the  base  or  root 
to  ease  the  approach;  even  this  is  done  only  in  a  few  instances. 
The  difference  between  the  hobbed  tooth  and  that  produced  by 
milling  is  seen  in  Fig.  16.  The  hobbed  tooth  is  shown  in  full; 
this  shape  was  traced  from  an  actual  hobbed  tooth,  photographed 
and  enlarged.  The  gear  had  twenty- 
one  teeth.  The  hob  used  was  corrected 
for  the  "  thinning  "  of  the  tooth  at  the 
point,  but  in  a  gear  of  this  diameter 
the  effect  would  not  show  to  any  great 
extent.  The  dotted  lines  are  drawn 
from  actual  milled  tooth  curves  and 
show  the  difference  between  the  two 
forms  of  teeth.  Attention  is  called  to 
the  fullness  of  the  milled  tooth  at  the 
root,  and  the  thinning  of  the  tooth  at 
the  point.  The  difference  would  be  greater  in  the  case  of  a 
twelve-  tooth  pinion. 

The  nlling-in  of  the  flank  of  the  tooth  is  not  done  to  any  rule 
based  on  a  proportion  to  the  number  of  teeth  in  the  gear.  The 
curve  selected  is  made  to  fill  in  the  space  at  the  root  to  just  clear 
the  corrected  rack  tooth.  Neither  is  the  thinning  of  the  tooth 
at  the  point  proportional  to  the  diameter  in  the  sense  that  the 
curve  of  the  hobbed  tooth  is.  Each  form  of  the  cutter  system  is 
made  and  varied  to  the  extent  necessary  for  smooth  action,  and 
the  curves  of  the  entire  system  cannot  be  produced  by  the  hobbing 
process  with  a  single  hob.  To  accurately  reproduce  the  form  of 
the  milled  tooth,  a  special  hob  would  be  necessary  for  each  num- 
ber of  teeth.  However,  a  close  approximation  may  be  obtained, 
within  a  narrow  range  of  teeth,  with  a  hob  generated  from  a 


FULL  LINES— BOBBED  TOOTH 


DOTTED  UNE8— MILLED  TOOTH 

Machinery 


Teeth 


158  SPUR   GEARING 

milled  tooth.  This  is  being  done  in  the  automobile  industry 
with  good  results.  The  necessity  for  interchangeability  makes 
the  duplication  of  the  milled  tooth  imperative  when  the  originals 
were  made  with  the  formed  cutter,  and  the  introduction  of  the 
hobbing  machine,  in  such  cases,  depends  on  the  successful  dupli- 
cation of  these  forms.  It  is  no  exceptional  thing  to  see  the  hob- 
bing process  used  in  conjunction  with  the  automatic  gear-cutter 
in  the  production  of  interchangeable  transmission  and  timing 
gears.  The  shapes  produced  by  the  standard  sets  of  cutters, 
from  a  rack  to  a  twelve-tooth  pinion,  cannot,  however,  be  gener- 
ated by  a  single  hob,  because  the  shapes  are  only  an  approxima- 
tion of  the  correct  curve.  The  gears  mentioned  above  as  being 
successfully  hobbed  are,  therefore,  when  milled,  cut  with  special 
cutters  for  each  number  of  teeth,  as  in  this  way  only  can  a  curve 
of  correct  shape  be  obtained. 

As  stated,  most  hobs  are  of  the  straight-sided  shape,  and  the 
tooth  hobbed  is  of  pure  involute  form.  In  gears  of  less  than 
thirty-two  teeth,  the  flank  is  undercut  to  a  considerable  extent. 
This  undercutting  does  not  involve  any  incorrect  action  in  the 
rolling  of  the  gears,  but  in  the  case  of  the  twelve-tooth  gear,  for 
example,  the  involute  is  cut  away  at  the  base  line  close  to  the 
pitch  line,  giving  but  a  line  contact  at  a  point  which  is  subjected 
to  heavy  wear.  This  eventually  develops  backlash.  The  teeth 
of  the  gears  also  come  into  action  with  a  degree  of  pressure  that 
is  continuous  throughout  the  time  of  contact;  this  results  in  a 
hammering  which  in  time  develops  into  a  humming  noise. 

Special  Hobs  for  Gear  Teeth.  —  To  overcome  these  objections 
a  hob  tooth  may  be  developed  to  generate  a  curve  which  will 
closely  resemble  that  of  the  formed  tooth.  Such  a  hob  tooth  is 
shown  in  Fig.  17.  Theoretically,  the  correction  for  interference 
or  undercutting  should  begin  at  a  point  located  above  the  pitch 
line  a  distance  as  determined  for  a  twelve-tooth  pinion  by  the 
expression: 

o-03 133  X-p 

in  which  N  =  number  of  teeth  in  the  smallest  gear  to  be  hobbed; 
P  =  diametral  pitch  of  gear. 


METHODS  OF  CUTTING  TEETH 


159 


However,  to  begin  the  correction  for  interference  at  this  point 
would  reduce  the  length  of  the  true  involute  and  result  in  too  full 
a  tooth,  causing  noisy  gears.  Therefore,  a  compromise  is  made 
and  the  correction  is  obtained  for  a  minimum  of  twenty-one 
teeth.  To  compensate  for  the  extra  fullness  of  the  tooth  at  the 
root,  the  point  of  the  tooth  is  thinned  down  in  proportion,  and 
this  is  done  by  leaving  the  tooth  of  the  hob  full  below  the  pitch 
line  by  striking  an  arc  from  a  center  on  the  pitch  line,  and  also 
employing  a  large  fillet  having  a  radius  equal  to  0.45  -f-  P  (see 
Fig.  17).  It  will  be  noticed  that  the  radius  of  the  arc  at  the  top 
of  the  hob  tooth  is  smaller  than  the  radius  at  the  bottom  of  the 
hob  tooth.  This  will  thin  the  tooth  of  the  gear  in  excess  of  the 
amount  necessary  to  clear  the  flank,  easing  the  action  and  elimi- 


0.154rP 


4+0.15-J-P 


js:V2^'-<rjt-      LINE  OF  RELIEF     P       I ' 
sNLl-—  -~/~~~          FOR  NON-INTEHFEREI 

— -%/^-45-P  | 

JL 


Machinery 


Fig.  17.    Hob  Tooth  designed  to  generate  the  Approximate  Shape  of 
a  ^-degree  Involute  Milled  Tooth 

nating  the  hammering  effect  due  to  the  theoretical  contact.  It 
will  be  seen  from  the  illustration  that  the  thinning  of  the  teeth 
does  not  affect  the  twelve-tooth  gear  to  any  appreciable  extent, 
but  is  gradually  increased  with  the  number  of  teeth.  The  fact 
that  a  twelve-tooth  gear  will  mesh  without  interference  at  the 
point  of  the  teeth  makes  the  thinning  unnecessary;  besides,  the 
small  pinions  are  usually  the  drivers. 

Fig.  1 8  shows  a  twenty-degree  pressure  angle  hob  tooth  with 
standard  addendum  and  corrections  for  non-interference.  The 
curve  of  the  tooth  begins  at  a  point  0.702  inch  from  the  pitch 
line,  in  the  case  of  a  one  diametral  pitch  tooth,  and  is  based  on 
non-interference  with  all  teeth  from  twelve  teeth  up. 

Fig.  19  shows  the  shape  of  the  hob  tooth  to  reproduce  the  stub 
teeth  of  the  gears  generated  on  the  Fellows  gear  shaper.  The 


160  SPUR   GEARING 

particular  tooth  in  the  figure  is  a  f-pitch  tooth,  and  the  propor- 
tions are  given  in  terms  of  the  pitch  numbers  so  as  to  be  easily 
applied  to  the  other  pitches;  thus  the  height  of  the  tooth  above 

the  pitch  line  is  stated  as :     -  +  •*—*  where  8  is  the  addendum 

8         8 

number  of  the  pitch  designation. 

The  shape  of  the  rack  or  hob  tooth  to  roll  with  the  gears  pro- 
duced by  the  gear  shaper  should  be  generated  from  the  cutter 
used.  The  Fellows  cutters  have  perfect  involutes  above  the 
base  line,  with  radial  flanks,  so  that  the  hob  tooth  would  be 
straight  only  a  distance  from  the  pitch  line  equal  to  0.0585  X 
N  -T-  P,  where  ./Vis  the  number  of  teeth  in  the  cutter;  in  most 
cases  the  cutter  would  have  more  than  seventeen  teeth  and  the 


Fig.  1 8.    Hob  Tooth  for  generating  a  2O-degree  Involute  Milled  Tooth 

hob  tooth  would  be  straight-sided  to  the  point.  In  this  system 
the  radial  flank  of  cutters  with  more  than  seventeen  teeth  does 
not  affect  the  shape  of  the  face  of  the  tooth,  as  the  involute  por- 
tion of  the  cutter  tooth  generates  a  pure  involute.  The  straight 
side  of  the  hob  tooth  should  extend  to  the  root  in  such  cases. 

To  reproduce  gears  of  some  standard  the  exact  shape  of  which 
is  not  known,  the  hob-tooth  shape  can  be  easily  generated  from 
the  gear  tooth  on  the  milling  machine,  as  will  be  explained  later. 

Generating  Hob-tooth  Shapes.  —  This  can  be  done  on  the 
universal  milling  machine,  or  on  the  plain  milling  machine  if  the 
screw  can  be  connected  up  with  the  worm  of  the  dividing  head, 
as  in  milling  spiral  work.  The  spindle  of  the  dividing  head  is  set 
vertical,  and  the  master  gear  or  templet  of  the  shape  it  is  desired 
to  produce  by  hobbing  is  mounted  on  an  arbor  in  the  spindle. 


METHODS  OF   CUTTING   TEETH 


161 


In  making  the  master  templets,  care  should  be  taken  to  produce 
the  correct  shape;  if  the  templet  is  not  true,  the  shape  generated 
will  not  be  accurate,  of  course. 

The  gearing  connecting  the  feed-screw  and  the  dividing-head 
must  be  for  a  lead  equal  to  the  circumference  of  the  pitch  circle 
of  the  gear  from  which  the  hob  templet  is  generated. 

To  provide  a  rest  on  which  the  tool  templet  to  be  laid  out  may 
be  clamped,  a  parallel  is  bolted  to  the  outer  arbor  support  so  as 
to  be  horizontal  and  parallel  with  the  milling  machine  table  and 
at  right  angles  to  the  machine  spindle.  The  rest  may  also  be  in 
the  form  of  an  angle  plate  clamped  to  the  face  of  the  column,  but 
the  former  type  is  the  most  desirable,  as  it  brings  the  work  in  a 
more  accessible  position. 

The  blank  templet  should  be  a  piece 
of  sheet  steel  about  one-sixteenth  inch 
thick,  one  edge  of  which  should  be 
straight  and  true  and  the  surfaces 
smooth  and  bright.  The  surface  to  be 
laid  out  can  be  given  a  coat  of  copper 
solution,  or,  still  better,  varnished  so 


Machinery 


that  the  lines  may  be  etched  deeper,  "fe2f  F°£wST<s£t « 
as  the  handling  in  working  out  the  Gear-Tooth 
shape  tends  to  obliterate  the  shallow  lines  in  the  thin  copper 
coat.  This  blank  templet  can  then  be  clamped  to  the  rest  in  a 
convenient  position. 

There  must  be  plenty  of  room  for  the  travel  of  the  gear, 
so  as  to  obtain  the  proper  amount  of  "  roll  "  to  generate  the 
shape  desired.  The  true  edge  of  the  plate  should  be  parallel 
with  the  rest  and  the  direction  of  the  movement  of  the  milling 
machine  table.  Adjust  the  knee  vertically  so  that  the  plate 
will  eome  up  under  the  gear  on  the  dividing  head  so  as  to 
just  clear  it;  the  saddle  can  then  be  adjusted  across  to  bring 
the  edge  of  the  plate  in  line  with  the  end  of  a  tooth  in  the  gear 
when  the  center  line  of  the  tooth  is  about  at  right  angles  to 
the  axis  of  the  feed-screw,  as  shown  in  Fig.  20.  In  this  way 
the  templet  is  set  to  the  proper  position  for  depth.  The  back- 
lash should  be  taken  up  by  turning  the  screw  in  the  direction 
in  which  it  is  to  be  used. 


162 


SPUR  GEARING 


Now  select  a  tooth  space  A,  Fig.  20,  as  the  one  to  be  used  in  the 
scribing  operation,  and  run  the  point  of  a  slim,  sharp  scriber 
along  the  outline  of  the  tooth  space,  scratching  the  line  on  the 
plate;  then  move  the  table  about  one-half  turn  of  the  lead-screw 
and  scribe  another  line,  and  repeat  the  operation  until  the  table 
has  been  moved  through  a  length  equal  to  three  times  the  circular 
pitch.  When  this  has  been  done  the  lines  on  the  templet  will 
resemble  that  in  Fig.  21.  The  lines  should  now  be  etched  in  and 
the  plate  polished. 


DIRECTION  OF  TABLE  TRAVEL 


SCRIBE  OUTLINE  OF  THE  SPACE  HERE. 
DOTTED  CURVE  SHOWS  SECOND  POSITION. 


Machinery 


Fig.  20.    View  showing  the  Relative  Position  of  Gear  and  Templet 

The  combined  lines  on  the  plate  will  be  seen  to  describe  the 
rack  tooth  shape  of  the  hob  teeth  in  a  clear-cut  manner,  if  the 
operation  has  been  carefully  carried  out.  If  the  gear  tooth 
from  which  the  lines  were  taken  is  theoretically  correct,  the  sides 
of  the  outline  on  the  plate  will  be  straight  a  greater  portion  of 
the  way  from  the  point  of  the  tooth  to  the  edge  of  the  plate;  the 
lines  diverge  from  the  straight  line  at  a  point  near  the  edge  of  the 
plate,  as  shown  by  the  dotted  lines  in  Fig.  21.  This  point  will 
be  found  to  be,  in  the  case  of  the  i^-degree  tooth,  at  a  distance 
from  the  pitch  line  of  0.03133  X  N  -r-  P,  where  N  is  the  number 
of  teeth  in  the  gear  and  P  the  diametral  pitch.  If  the  hob  to  be 


METHODS  OF   CUTTING  TEETH 


I63 


made  from  this  form  is  to  be  used  for  N  teeth  or  less,  the  shape 
of  the  templet  will  be  correct,  but  if  the  hob  is  to  cut  gears  of 
a  larger  number  of  teeth,  the  straight  portion  of  the  tooth 
must  be  carried  down  to  the  edge  of  the  plate  in  order  that  the 
teeth  of  the  larger  gears  will  not  be  cut  away  too  much  at  the 
points. 

In  making  a  templet  in  this  way  for  any  other  shape  than  for 
gears,  it  should  be  cut  to  the  lines  on  the  plate,  as  no  correction 
can  be  intelligently  made  in  those  cases.  Some  success  has  been 
made  in  the  layout  of  templets  for  a  hob  tooth  for  gears  of  a 
limited  range  of  teeth,  by  using  the  space  below  the  pitch  line 


Machinery 


Fig.  21.    Lines  scribed  on  the  Hob  Tooth  Templet  from  a  Hobbed  Gear 

of  the  smallest  gear  in  the  set  and  the  space  above  the  pitch  line 
of  the  largest  gear  in  the  set  as  the  shape  in  generating  the  hob 
tooth  templet.  This  is  of  value  in  generating  a  hob-tooth  shape 
to  reproduce  a  set  of  gears  milled  with  formed  cutters.  How- 
ever, the  best  and  easiest  method  is  to  take  the  smallest  gear  in 
the  set  as  the  one  from  which  to  generate,  and  prolong  the 
straight  portion  of  the  hob  tooth  to  the  edge  of  the  plate,  easing 
the  side  at  A,  to  point  the  teeth  slightly.  The  teeth  of  the  hob 
are  generally  made  with  an  extra  clearance  at  the  bottom,  as 
shown.  This  is  a  matter  on  which  authorities  differ,  some  pre- 
ferring to  have  the  hob  cut  the  top  of  the  teeth,  to  make  the  teeth 
of  standard  length  if  the  blanks  should  be  over  size;  however, 


164  SPUR   GEARING 

it  is  also  practiced  to  make  the  tooth  the  same  length  both 
above  and  below  the  pitch  line  as  in  Figs.  17  and  18. 

If  the  form  is  for  a  generated  gear  and  results  in  the  straight- 
sided  tool  in  Fig.  21,  all  that  is  necessary  is  to  measure  the  angle 
and  make  a  thread  tool  that  will  cut  a  thread  of  this  section. 
Should  the  shape  turn  out  to  be  a  compound  of  curves,  as  will 
be  the  case  in  reproducing  the  milled  tooth,  the  templet  should 
be  filed  out  to  the  line's,  making  a  female  gage  to  which  a  planing 
tool  is  made,  the  planing  tool  being  a  duplicate  of  the  hob-tooth 
shape.  The  threading  tool  is  planed  up  with  this  tool.  In 
making  the  thread  tool,  it  is  not  usual  to  make  it  of  female  shape, 
that  is,  like  the  templet,  but  pointed  as  usual,  planing  the  sides 
with  the  opposite  sides  of  the  planing  tool.  The  proper  cor- 
rections should  be  made  in  the  thread  tool  to  correspond  to  the 
angle  of  the  thread,  and  the  setting  of  the  tool  and  the  fluting 
of  the  hob,  whether  it  is  gashed  parallel  to  the  axis  or  normal  to 
the  thread  helix. 

Making  a  Master  Planing  Tool  for  a  Hob.  —  A  master 
planing  tool  can  be  made  in  the  following  manner,  without  the 
use  of  the  scribed  line  templet.  It  is  necessary  to  have  a  univer- 
sal milling  attachment  for  the  milling  machine.  The  spindle 
of  the.  attachment  is  set  in  the  horizontal  position  with  the  axis 
parallel  with  the  direction  of  the  table  movement.  A  fly-tool 
holder  is  then  placed  in  the  spindle,  in  which  the  blank  planing 
tool  is  to  be  held.  This  tool  should  be  roughly  formed  to  the 
shape  to  which  it  is  to  be  finished.  The  top  of  the  tool  should 
be  radial,  that  is,  it  should  be  in  the  plane  of  the  center  of  the 
spindle.  The  gear  or  other  master  templet  that  it  is  desired  to 
duplicate  by  hobbing  must  be  hardened  and  ground  to  a  cutting 
edge  on  one  face,  preferably  the  top  face  when  mounted  in  the 
spindle  of  the  dividing  head,  so  that  the  pressure  of  the  cut  will 
be  downward.  The  knee  should  be  adjusted  to  bring  the  ground 
face  of  the  gear  to  the  level  of  the  center  of  the  spindle  of  the 
attachment.  The  dividing  head  and  the  table  screw  are  con- 
nected in  the  same  way  as  previously  described,  but  in  this  case 
the  power  feed  can  be  used  and  the  saddle  can  be  fed  in  to  depth 
as  needed,  care  being  taken  to  use  the  power  feed,  in  generating 


METHODS  OF  CUTTING  TEETH  165 

the  tool,  only  in  one  direction,  as  the  backlash  in  the  gears  and 
screw  will  throw  the  tool  and  dividing  head  out  of  relative  posi- 
tion if  used  in  the  opposite  direction.  As  many  cuts  can  be  taken 
as  required  to  obtain  a  tool  of  the  correct  shape. 

If  the  tool  is  to  be  used  in  making  more  than  one  threading 
tool,  as  might  be  the  case  in  many  instances,  the  planing  tool 
can  be  made  in  the  shape  of  a  circular  tool  which  can  be  ground 
indefinitely  without  losing  its  shape.  In  this  case  the  fly-tool 
holder  would  give  place  to  the  standard  milling  machine  arbor. 
This  method  is  the  most  accurate  way  of  making  the  master 
planing  tool,  and  where  the  universal  milling  attachment  is 
available,  it  should  be  used  when  accurate  results  are  desired. 
It  eliminates  the  human  element  and  the  amount  of  skill  required 
in  making  the  master  templet.  The  inaccuracy  of  the  machine 
is  the  only  element  that  is  likely  to  cause  error. 

One  point  that  is  likely  to  cause  difficulty  is  the  relation  of  the 
generated  tool  shape  to  the  thread  shape,  as  it  appears  in  the 
normal  section  of  the  hob  tooth.  The  simple  fact  is  that  the 
master  tool  shape,  as  generated  by  the  direct  method  of  making 
the  master  planing  tool,  or  the  shape  as  outlined  on  the  hob  tooth 
templet  in  the  first  method,  is  the  shape  of  the  cross-section  of 
the  hob  thread  on  a  plane  normal  to  the  hob  thread  helix.  This 
relation  should  be  kept  in  mind  throughout  the  process  of  making 
the  tools  and  hob.  This  statement  also  clears  any  haziness  re- 
garding the  question  of  the  lead,  as  in  single-threaded  hobs  this 
must  be  such  that  the  normal  pitch  of  the  thread  is  equal  to  the 
circular  pitch  of  the  teeth  hobbed.  In  the  case  of  hobs  of  small 
thread  angles,  the  normal  and  axial  leads  are  practically  the  same, 
and  may  be  treated  as  such  in  cases  where  the  angle  is  less  than 
2  degrees  and  the  pitch  less  than  ^  inch;  an  error  of  more  than 
0.00025  inch  should  not  be  exceeded  in  any  case.  The  effect  of 
the  error  is  apparent  in  the  case  of  a  6  diametral  pitch  hob  3 
inches  in  diameter,  when  the  axial  lead  is  taken  as  the  circular 
pitch  of  the  teeth,  as  it  results  in  an  error  of  more  than  one-half 
degree  in  the  pressure  angle  of  the  hobbed  tooth. 

Only  in  extreme  cases  should  the  angle  of  the  hob  thread  be 
more  than  ten  degrees.  Hobs  with  greater  angles  than  this  are 


l66  SPUR  GEARING 

difficult  to  make  and  use.  In  hobs  of  long  lead  the  diameter 
should  be  proportioned  so  as  to  obtain  a  reasonable  angle  of 
thread.  However,  the  extreme  in  diameters  is  as  bad  as  the 
steep  angles,  and  in  cases  where  the  two  extremes  are  met  a 
compromise  is  the  only  solution. 

The  method  used  in  laying  out  the  tooth  shapes  on  the  draw- 
ing-board is  an  interesting  study,  but  the  method  of  generating 
the  tool  as  described  is  the  most  useful,  and  can  be  relied  on  for 
accurate  results;  this  is  not  the  case  with  the  drawing-board 
method  which  is  of  value  only  as  a  means  of  getting  an  approxi- 
mate shape. 


CHAPTER  IX 

METHODS    OF   PRODUCTION   AND    HEAT-TREATMENT 

OF    GEARS 

Processes  in  Production  of  Automobile  Transmission  Gears.  — 

One  of  the  most  important  problems  in  modern  automobile 
construction,  and  one  which  has  received  a  great  deal  of  atten- 
tion from  mechanical  engineers  during  the  past  few  years,  is 
that  of  the  quiet  working  of  the  running  parts.  Next  to  the 
engine  itself,  the  gears  have  proved  the  greatest  offenders  in 
making  noise.  Therefore,  the  demand  for  gears  which  are 
accurate,  interchangeable  and  silent,  together  with  the  neces- 
sity for  producing  them  both  rapidly  and  at  a  low  cost,  has 
caused  a  great  deal  of  attention  to  be  devoted  to  the  various 
processes,  tools  and  appliances  whereby  that  demand  may  be 
satisfied. 

We  thus  find  that  there  are  being  placed  on  the  market  an 
increasing  number  of  machine  tools,  steels  and  carbonizing 
materials,  each  of  which  claims  some  advantage  over  its  prede- 
cessors, such  as  increased  output,  greater  simplicity,  superior 
generating  features,  and  better  hardening  results.  It  is  the 
intention,  however,  to  deal  here  chiefly  with  the  processes  of 
manufacturing  gear-box  gears  by  means  of  a  complete  equip- 
ment of  gages,  tools,  jigs,  etc.,  with  the  object  of  insuring  inter- 
changeability.  To  a  very  large  extent  fitting  is  thus  dispensed 
with.  After  the  final  machining  operation  has  been  performed, 
the  parts  should  be  ready  to  be  assembled.  When  a  duplicate 
part  is  wanted  it  can  be  supplied  from  stock,  as  the  methods  here 
to  be  described  insure  that  it  will  fit  into  its  correct  position  with- 
out trouble.  These  methods  were  outlined  in  a  paper  read  by 
Mr.  W.  Betterton  before  a  branch  of  the  Institute  of  Automobile 
Engineers. 

There  is  probably  no  part  of  an  automobile  that  is  subjected 

167 


1 68  SPUR   GEARING 

to  greater  use  —  and  abuse  —  than  the  gears,  especially  the 
gear-box  gears.  Carrying,  as  they  do,  practically  all  the  power 
developed  by  the  engine,  and  receiving  at  the  hands  of  a  care- 
less driver  the  strains  imparted  by  suddenly  applied  load  or 
by  rapid  changes,  it  is  absolutely  necessary  that  the  gears  be 
made  of  the  highest  grade  materials,  and  that  the  very  greatest 
care  and  the  best  workmanship  should  be  bestowed  upon  them. 
As  the  saving  in  weight  is  an  important  factor  in  the  design  of 
the  transmission,  the  gears  must  be  made  as  small  and  as  light 
as  possible,  and  yet  be  sufficiently  strong  to  carry  suddenly 
applied  loads  with  no  danger  of  breaking.  Owing  to  the  methods 
by  which  the  speeds  are  changed,  and  the  clashing  and  bruising 
to  which  the  gears  are  thus  subjected,  the  transmission  mech- 
anism must  be  made  of  material  which  is  both  hard  and  tough. 
Different  kinds  of  steel  have  been  used,  and  each  has  been  treated 
by  various  methods  in  the  attempt  to  discover  the  perfect  gear 
material.  Although  this  has  not  yet  been  found,  so  much  prog- 
ress has  already  been  made  that  the  transmission  gear  of  a 
modern  well-made  automobile,  when  carefully  handled,  will  last 
nearly  as  long  as  the  car  itself. 

Steels  used  for  Automobile  Gears.  —  Of  the  various  kinds  of 
steel  which  have  hitherto  been  employed,  nickel,  nickel-chromium 
and  chrome- vanadium  steels  seem  to  have  more  advocates  than 
any  others.  In  most  factories  the  gears  are  casehardened,  and 
it  is  this  class  of  gear  which  will  be  dealt  with  in  the  following. 
Gears  treated  in  this  way  have  been  taken  out  of  cars  which  have 
been  run  many  thousands  of  miles,  and  in  some  instances  the 
original  tool-marks  on  the  face  of  the  teeth  were  still  visible. 

The  processes  in  the  manufacture  of  low-carbon  nickel-steel 
casehardened  gears,  such  as  the  finished  gears  shown  by  Figs, 
i,  2  and  8,  will  be  described  in  the  following.  The  various 
processes  will  be  explained  in  the  order  in  which  they  are  per- 
formed. The  composition  of  nickel  steel,  suitable  for  high- 
speed gears,  is  as  follows:  Carbon,  0.20  per  cent;  manganese, 
0.65  per  cent;  silicon,  not  exceeding  0.20  per  cent;  phosphorus, 
not  exceeding  0.04  per  cent;  sulphur,  not  exceeding  0.04  per 
cent;  and  nickel,  3.50  per  cent.  Steel  with  3.50  per  cent  nickel 


MACHINING  AND   HEAT-TREATMENT 


169 


rolls  and  forges  well,  and,  when  hardened,  the  ratio  of  the  elastic 
limit  to  the  ultimate  strength  is  very  great.  The  influence  of 
nickel  on  steel  is  that  it  increases  the  tensile  strength  and  the 
elastic  limit. 


7777/s 


Fig.ii 


Fig.  3 


Fig.  1 


Fig.  5 


, _| 


Fig.  7 


Fig.  6 


Machinery 


Figs,  i  to  7.    Automobile  Transmission  Gears  at  Various  Stages 
of  Completion 

Nickel  steel  of  the  composition  just  mentioned  should  have 
an  elastic  limit,  after  treatment,  of  about  30  tons  per  square  inch. 
The  influence  of  silicon  on  the  results  of  quenching  is  similar  in 
many  ways  to  that  of  carbon.  It  is  dependent  on  the  co-existing 
amount  of  carbon  and  manganese,  and  it  is  difficult  to  obtain 


1 70  SPUR   GEARING 

silicon  in  steel  without  the  presence  of  manganese.  Silicon  ap- 
pears to  increase  the  tensile  strength  and  diminish  the  ductility; 
but  for  various  reasons  it  is  generally  considered  objectionable. 

Phosphorus  is  the  least  desirable  element  in  steel,  but  up  to 
one  per  cent  it  appears  to  increase  the  tensile  strength.  Sul- 
phur tends  to  produce  hot-shortness  and  difficulty  in  working, 
but  in  the  presence  of  manganese  the  effect  is  diminished. 

The  Gear  Blanks.  —  Blanks  for  gears  of  the  type  shown  in 
Fig.  i  should  be  cut  from  the  bar,  since  it  has  been  proved  that 
steel  is  not  improved  by  drop  forging,  although  some  steels  are 
less  sensitive  to  injury  than  others.  An  investigation  of  drop- 
forged  and  bar-cut  nickel-steel  gears,  details  of  which  were  given 
in  a  paper  read  by  Mr.  John  A.  Mathews  before  the  Franklin 
Institute,  showed  that  under  static  tests  the  bar-cut  gears  were 
fully  25  per  cent  stronger,  and  also  that  the  resistance  to  shock 
was  greater.  The  gears  shown  in  Fig.  8  should  be  made  from 
a  drop  forging,  as  shown  by  Fig.  3,  although  when  only  small 
quantities  are  required,  it  would  not  pay  to  make  dies.  In  this 
case  ordinary  forgings  should  be  considered.  Gear  blanks  should 
be  annealed  previous  to  machining. 

The  reasons  for  leaving  so  much  extra  metal  will  be  explained 
in  the  order  in  which  they  concern  the  various  operations  neces- 
sary in  the  attempt  to  get  a  perfect  gear  —  an  end  which,  it  is 
needless  to  say,  is  seldom,  if  ever,  attained.  In  the  case  of  a 
gear  as  shown  in  Fig.  2,  made  from  a  bar,  it  is  not  necessary,  for 
reasons  which  will  be  seen  later,  to  leave  any  extra  metal.  Much 
of  the  trouble  due  to  distortion  in  the  heat-treatment  is  caused 
by  the  forging  operations  being  done  at  too  low  a  temperature, 
in  which  case  the  metal  does  not  have  a  chance  to  flow  properly, 
but  is  merely  forced  into  shape  by  the  die.  This  sets  up  internal 
strains  that  will  be  released  when  the  part  is  annealed. 

Rough-turning.  —  The  first  operation  is  to  rough-turn  the  part 
all  over  for  the  purpose  of  removing  the  outer  skin,  previous  to 
the  second  annealing,  leaving  a  one-sixteenth  inch  case  on  the 
parts  required  to  be  hardened,  such  as  the  top  diameter  of  the 
gear,  and  the  sides  of  the  teeth.  For  the  bore  a  one-eighth  inch 
allowance  is  required,  in  gears  where  the  hole  is  to  be  a  running 


MACHINING  AND   HEAT-TREATMENT  171 

fit,  or  castellated,  and  has  to  be  hard.  A  one-quarter-inch 
allowance,  however,  is  necessary  when  the  gear  has  to  be  bolted 
to  a  center,  or  to  another  gear,  in  which  case  the  bore  need  not 
be  hard,  as  it  is  only  used  for  locating  the  gear  centrally.  In 
rough-turning,  allowance  must  be  made  for  the  extra  metal,  the 
gears,  after  machining,  appearing  as  shown  in  Figs.  4  and  5. 

Annealing.  —  When  making  gears  it  is,  of  course,  necessary  to 
have  the  steel  carefully  and  uniformly  annealed.  The  process 
of  annealing  is  one  of  great  importance,  and  is  better  performed 
in  a  specially  designed  sealed  furnace,  constructed  as  a  muffle,  so 
that  the  required  heat  is  obtained  uniformly  by  radiation,  with- 
out any  flame  to  impinge  on  the  steel.  In  addition  to  softening 
the  steel,  and  making  it  easy  to  machine,  annealing  has  the  effect 
of  bringing  it  to  a  more  homogeneous  condition  by  eliminating 
the  molecular  strains  which  are  set  up  by  rolling,  hammering  and 
stamping.  Hence,  when  the  steel  is  heated  preparatory  to  hard- 
ening, equal  expansion  should  follow,  and  also  equal  contraction 
when  cooled. 

It  will  thus  be  seen  that  should  the  steel  not  be  annealed  uni- 
formly throughout,  the  risks  of  warping  when  hardening  are 
considerably  increased.  The  object  of  rough-machining  is  to 
break  down  the  scale  preparatory  to  the  second  annealing,  and 
as  the  strains  set  up  by  rough-machining  are  released  by  the 
second  annealing,  the  metal  is  then  in  as  normal  a  condition  as 
possible.  At  the  present  time  there  are  many  compounds  used 
for  annealing.  A  few  years  ago,  the  ashes  from  the  forge  were 
considered  sufficient  for  properly  annealing  steel,  but  to-day  many 
special  preparations  are  manufactured  and  sold  for  the  purpose. 

The  more  common  materials  used  are  powdered  charcoal, 
charred  leather  and  hydro-carbonated  bone-black.  These  same 
materials  are  used  for  carbonizing,  but  after  having  been  used 
once  they  are  of  very  little  use  for  that  purpose.  However, 
their  use  for  annealing  has  the  additional  merit  of  economy, 
because  they  can  be  used  repeatedly,  adding  each  time  a  little 
that  has  only  been  used  for  the  carbonizing  process.  Air- 
slaked  lime  may  also  be  used  for  this  process.  The  piece  to 
be  annealed  is  usually  packed  in  a  wrought-iron  box,  using  one 


172 


SPUR   GEARING 


of  the  previously  mentioned  materials,  or  combinations  of  them, 
for  the  packing.  The  whole  is  then  heated  to  the  proper  tem- 
perature, which  is  about  1760  degrees  F.  In  the  case  of  the 
gears  in  question,  this  temperature  should  be  maintained  for 
one  hour.  The  box  may  then  be  set  aside  with  the  cover  on 
in  order  to  cool  down  to  atmospheric  temperature,  or  it  may 
cool  off  with  the  furnace.  It  should  be  noted  that  the  annealing 
temperature  ought  always  to  be  higher  than  that  for  carbon- 
izing. For  all  kinds  of  steel  and  for  all  grades  of  annealing,  the 
slow-cooling  furnace  gives  the  best  results,  because  the  tem- 
perature can  easily  be  raised  to  the  right  point,  kept  there  as 
long  as  necessary,  and  then  regulated  to  cool  down  as  slowly 
as  desired.  Gas,  oil,  or  electric  furnaces  are,  of  course,  the 
easiest  to  regulate. 

Finish-boring  and  Broaching.  —  Gears  to  be  broached,  as 
shown  in  Fig.  8,  should  be  finish-bored  with  an  allowance  of 
0.015  mcn  f°r  grinding  after  hardening,  and  faced  on  one  end 
true  with  the  bore  to  take  the  thrust  of  the  broaching  on  a  La- 
pointe  or  similar  broaching  machine.  The  broaches  should  be 
made  of  carbon  steel,  oil  hardened,  tempered  and  ground,  and 
should  be  specially  treated.  It  is  necessary  to  treat  the  steel 
with  some  carbonaceous  material  until  it  will  harden  in  oil,  as  it 
is  well  known  that  steel  hardened  in  oil  is  less  likely  to  spring 
than  if  hardened  in  water.  The  tendency  for  steel  to  crack  is 
almost  eliminated  and  it  has  a  maximum  of  toughness,  unless, 
of  course,  the  steel  has  been  improperly  treated  in  the  fire. 
The  special  treatment  consists  essentially  in  supplying  the  sur- 
face of  the  steel  with  an  additional  amount  of  carbon  by  some 
material  that  will  not  injure  the  steel.  No  form  of  bone 
should  be  used  on  tool  steel  for  this  process,  as  bone  contains  a 
high  percentage  of  phosphorus,  and  the  effect  of  this  is  to  make 
the  steel  weak  and  brittle.  Charred  leather  gives  the  best 
results. 

The  over-all  length  of  the  roughing  broaches,  suitable  for  the 
gears  shown  in  Fig.  8,  would  be  about  48  inches,  with  the  last 
four  teeth  parallel  on  each  broach.  The  pitch  depends  upon 
the  length  of  the  work  being  broached,  but  may  be  about  f  inch, 


MACHINING  AND   HEAT-TREATMENT 


173 


as  an  average.  The  teeth  should  never  be  cut  spiral,  as  this 
tends  to  twist  the  broach  while  in  operation  and,  consequently, 
the  castellation  would  not  be  true.  To  produce  an  accurate 
castellated  hole,  as  shown  in  Fig.  8,  in  nickel  steel,  it  is  necessary 
to  use  two  roughing  broaches  and  one  finishing  broach.  The 
latter  will  be  about  36  inches  long,  and  will  remove  only  from 
o.ooi  to  0.002  inch  of  material.  There  should  be  a  parallel  por- 
tion of  about  10  inches  at  the  end  of  the  third  or  finishing 
broach.  On  the  first  broach,  the  pilot  should  be  round  and  of 
the  same  diameter  as  the  hole  in  the  gear  to  be  broached.  On 


Machinery 


Fig.  8.    Completed  Gear,  shown  in  Progress  of  Evolution  in 
Figs.  3,  5  and  7 

the  following  broaches,  the  pilot  should  be  a  sliding  fit  in  the 
hole  produced  by  the  previous  broach,  and  should,  at  the  same 
time,  be  located  from  the  castellations. 

The  finishing  broach  should  have  a  piece  at  the  rear  end, 
about  3  inches  long,  of  the  exact  size  of  the  castellated  shaft  on 
which  the  gear  is  to  be  fitted  when  finished.  This  will  act  as 
a  burnisher.  The  broaches  are  pulled  through  the  work,  which 
is  quite  a  good  feature,  as  it  tends  to  keep  them  straight  while 
in  operation.  The  cutting  portion  being  so  long  enables  the 
operation  to  be  performed  without  previous  rough-slotting, 
which  is  required  when  the  gears  are  drifted  on  the  power  press. 
After  broaching,  the  gears  should  be  turned. 


174  SPUR   GEARING 

Finish-turning.  —  Castellated  gears  should  be  turned  on  a 
mandrel,  locating  from  the  castellations.  The  parts  required  to 
be  hard,  such  as  the  top  diameter  of  the  gears,  which  should  have 
an  allowance  of  0.005  mcn  f°r  gear-cutting  purposes,  and  the 
sides  of  the  teeth  and  fork  groove  are  then  finished,  leaving 
the  gear  as  shown  in  Fig.  5.  In  the  case  of  the  gear  in  Fig.  i, 
the  top  diameter  should  be  finish-turned  with  an  allowance  of 
0.005  mcn>  and  the  sides  of  the  teeth  and  the  bore  with  an  allow- 
ance of  0.125  inch,  so  that  the  latter  can  be  bored  out  again 
after  carbonizing,  which  would  leave  the  hole  soft,  as  it  is  not 
required  to  be  hard.  The  finish-turning  leaves  the  gear  as 
shown  in  Fig.  4.  Gears  such  as  shown  in  Fig.  2  can  be  finish- 
turned  and  bored  complete  at  this  operation. 

Cutting  the  Teeth  —  Roughing  Operation.  —  One  of  the  most 
important  operations  is  the  cutting  of  the  teeth  after  annealing. 
The  method  to  be  described  is  at  present  in  practice  and  gives 
very  successful  results.  The  teeth  should  be  roughed  out  on 
the  hobbing  machine,  and,  where  possible,  several  gears  should 
be  placed  on  the  work-arbor  at  one  setting.  A  great  deal  of 
time  is  saved  by  placing  several  gears  on  the  work-arbor,  which 
should  always  be  steadied  at  the  top.  Suppose  six  are  to  be  cut 
at  one  setting;  this  would  mean  that  the  hob  would  only  have 
to  travel  into  and  clear  the  work  once  instead  of  six  times,  which 
would  be  necessary  if  they  were  cut  singly.  When  putting 
several  gears  on  the  arbor,  they  must  be  faced  exceedingly  true, 
or  the  arbor  will  be  bent.  This  is  especially  true  when  the  hole 
is  small  and  the  gears  large  in  diameter.  Plenty  of  lubricant 
should  be  applied  in  this  operation,  oil  being  most  commonly 
used.  The  hob  should  be  made  of  high-speed  steel,  a  six-pitch 
hob  being  3  inches  in  diameter,  and  a  five-pitch  hob,  3^  inches  in 
diameter.  On  account  of  the  accuracy  required,  single-threaded 
hobs  are  preferable.  For  the  roughing  operation  a  cutting  speed 
of  sixty  feet  per  minute  of  the  hob,  and  0.020  inch  feed  per 
revolution,  are  considered  good  practice  for  a  nickel-steel  six- 
pitch  gear. 

Cutting  the  Teeth  —  Finishing  Operation.  —  For  the  finishing 
operation  at  least  o.oio  inch  should  be  allowed  for  a  six-pitch 


MACHINING  AND   HEAT-TREATMENT  175 

gear  and  other  pitches  in  proportion.  If  the  finishing  cut  is 
merely  a  scraping  cut  and  not  enough  stock  is  removed  to  let 
the  cutter  get  a  real  chip,  the  cutter  may  glaze  over  the  work, 
especially  if  the  cutter  and  the  work-arbor  are  not  held  rigidly. 
The  gears  should  be  finished  one  at  a  time,  excepting  plate  gears, 
in  which  case  several  can  be  placed  on  the  arbor  at  one  setting. 
For  this  purpose  a  gear  shaper  should  be  used,  because  the  cutter 
can  be  made  far  more  accurate  than  a  hob  or  a  rotary  cutter. 
The  teeth  of  this  cutter  can  be  ground  after  hardening,  and  this 
corrects  any  inaccuracies  that  may  have  crept  in.  On  the  cutter- 
arbor,  at  the  back  of  the  gear-cutter,  should  be  placed  a  round 
disk  made  of  high-speed  steel,  hardened,  ground  and  backed  off, 
which  will  act  as  a  shaving  tool  and  will  take  off  the  0.005  mcn 
left  on  the  top  diameter,  as  previously  stated.  This  will  make 
the  outside  diameter  true  with  the  pitch  line.  The  teeth  should 
be  cut  about  o.ooi  inch  thin  at  the  pitch  line  to  produce  a  running 
fit.  The  gears  should  now  be  tested  for  center  distance,  for 
which  purpose  a  plate  with  two  pins  for  the  bores,  set  at  the 
correct  center  distance,  should  be  used. 

Rounding  the  Ends  of  the  Teeth.  —  The  tooth-rounding  should 
be  performed  on  an  automatic  tooth-rounding  machine.  Several 
gears  should  be  mounted  on  the  same  arbor,  and  the  teeth  of  the 
wheels  may  be  rounded  in  succession  at  one  setting.  No  doubt 
most  readers  are  quite  familiar  with  the  reason  for  this  operation, 
which,  therefore,  requires  little  explanation,  unless  it  be  to  say 
that  it  is  done  to  facilitate  the  changing  of  the  gears,  and  also 
to  prevent  the  teeth  from  being  chipped  when  engaging,  which 
would  occur  if  the  corners  were  left  square.  The  sides  of  the 
teeth  which  are  not  rounded  should  be  fraised  before  carbonizing, 
which  is  the  next  operation. 

The  Carbonizing  Process.  —  In  carbonizing,  great  care  should 
be  taken,  as,  to  a  very  great  extent,  the  life  of  the  gear  depends 
on  this  process.  The  result  of  the  process  is  determined  by  four 
factors,  namely:  the  nature  of  the  steel;  the  nature  of  the  car- 
bonizing materials;  the  temperature  of  the  carbonizing  furnace; 
and  the  time  taken  by  the  process.  The  carbonizers  in  general 
use  at  the  present  time  are  animal  charcoal,  hydro-carbonated 


1 76  SPUR   GEARING 

bone-black,  charred  leather,  and  a  few  other  compositions  sold 
under  various  names.  Owing  to  the  various  conditions  under 
which  the  operation  is  carried  out,  experience  must  largely  guide 
the  operator.  Theoretically,  the  perfect  carbonizer  should  be 
a  simple  form  of  carbon,  and  charred  leather  gives  very  satis- 
factory results.  Care  should  be  taken  to  avoid  poorly  charred 
leather,  or  that  made  from  old  boots,  belting,  etc.  Good  charred 
leather  should  contain  about  88  per  cent  of  carbonizing  matter. 

As  it  is  essential  that  the  core  of  the  gears  should  be  left  soft 
in  order  to  withstand  the  high  speed  and  sudden  shocks  to 
which  they  are  subjected,  the  carbon  content  in  the  core  should 
be  low.  For  this  reason  preference  is  given  to  0.20  per  cent 
carbon  steel.  The  carbonizing  pots  are  made  from  both  cast 
and  wrought  iron;  the  former  are  cheaper  in  first  cost,  but  the 
latter  bear  reheating  so  many  times  that  they  are  really  cheaper 
in  the  end.  The  carbonizer  having  been  thoroughly  dried  and 
reduced  to  a  fine  powder,  a  layer  of  not  less  than  i^  inch  in  depth 
is  placed  in  the  carbonizing  pot  and  well  pressed  down.  Upon 
this  are  placed  the  articles  to  be  treated.  Care  must  be  taken 
to  have  sufficient  space  all  around  each  piece  so  as  to  prevent 
them  from  touching  each  other  or  the  walls  of  the  pot.  About 
ij  inch  is  sufficient.  Another  layer  of  carbonizing  material  is 
then  put  in  and  well  pressed  down,  care  being  taken  not  to  dis- 
place any  of  the  gears.  The  process  is  then  continued  until  the 
pot  is  full,  finishing  with  a  layer  of  about  ij  inch  at  the  top. 

The  object  in  view  is  to  make  the  contents  of  the  pot  as  com- 
pact as  possible,  consistent  with  a  sufficiency  of  carbonizer  in 
contact  with  the  gears.  The  more  solidly  the  pot  is  packed, 
the  more  complete  the  exclusion  of  air.  The  lid  is  then  put  on, 
and  the  joint  luted  with  clay  all  around.  The  pot  should  be 
placed  in  a  furnace  similar  to  that  used  for  annealing,  and  heated 
to  about  1700  degrees  F.,  which  heat  should  be  maintained  con- 
stant for  from  six  to  ten  hours.  The  length  of  time  occupied  is 
regulated  by  the  depth  of  casing  required,  which  should  be  about 
three-sixty-fourths  inch,  and  also  by  the  dimensions  of  the  gears. 
At  the  close  of  the  carbonizing  period,  the  pot  is  withdrawn  and 
put  in  a  dry  place  where  it  is  allowed  to  cool  to  atmospheric 


MACHINING  AND   HEAT-TREATMENT  177 

temperature.  It  is  then  opened,  the  articles  are  taken  out,  and 
the  process  is  completed  by  brushing  to  remove  all  adhering 
matter. 

It  may  be  noted  here  that  the  case  should  only  be  deep  enough 
to  resist  wear  and  battering.  If  the  case  is  so  deep  as  to  form  a 
considerable  part  of  the  cross-section  of  the  teeth,  the  teeth  may 
break  unless  the  case  is  considerably  tempered. 

Turning  Operations  Preceding  Hardening.  —  The  next  opera- 
tion is  to  turn  out  the  carbon  from  the  parts  required  to  be  soft. 
In  the  case  of  the  plate  gear  shown  in  Fig.  i ,  which  is  to  be  bolted 
to  another  gear  or  center,  as  previously  stated,  the  hole  need  not 
be  hard.  For  this  reason,  it  was  left  with  an  allowance  of  f  inch 
at  the  previous  turning  operation.  It  should  now  be  bored  with 
an  allowance  of  0.015  inch  for  grinding  after  hardening,  and  faced 
down  on  both  sides.  This  refers  to  both  plate  and  castellated 
gears  as  shown  in  Figs.  6  and  7.  The  boring  operations  should 
be  performed  while  the  gear  is  held  in  a  collet-chuck,  locating 
from  the  top  of  the  teeth,  which  were  trued  up  with  the  pitch  line 
by  the  shaving  tool  used  when  cutting  the  teeth,  as  previously 
explained.  This  method  has  been  found  to  be  more  efficient 
than  locating  with  balls  or  rollers  on  the  pitch  line.  The  pene- 
tration of  carbon  being  only  about  three-sixty-fourths  inch  it  is 
now  removed  from  the  parts  which  have  just  been  turned.  Con- 
sequently these  parts  will  not  be  hardened.  The  excess  metal 
is  left  as  shown  in  Figs.  6  and  7  until  after  hardening,  in  order  to 
prevent  warping,  which  would  undoubtedly  happen  if  the  gears 
were  finished  as  shown  by  Figs,  i  and  8. 

Casehardening  and  Oil  Tempering.  —  Steel  of  the  composi- 
tion mentioned  can  be  hardened  as  follows:  Heat  from  1450  to 
1525  degrees  F.  and  quench  in  water;  reheat  from  about  1400 
to  1450  degrees  F.  and  quench  in  water.  The  reheating  must 
be  conducted  at  the  lowest  possible  temperature  at  which  the 
steel  will  harden.  It  will  be  found  that  this  is  sometimes  as  low 
as  1300  degrees  F.  Then,  as  a  safeguard,  reheat  to  a  temperature 
of  between  250  and  500  degrees  F.,  in  accordance  with  the  require- 
ments of  the  case,  and  cool  slowly  in  oil.  Parts  of  intricate  shape, 
such  as  the  gears  dealt  with,  having  sudden  changes  of  thickness, 


178  SPUR  GEARING 

sharp  corners  and  the  like,  should  always  be  tempered  or  drawn 
in  order  to  relieve  internal  strains. 

Another  method  of  procedure  is  as  follows:  Heat  from  1450 
to  1525  degrees  F.,  and  quench  in  hot  brine.  Reheat  from  1450 
to  1525  degrees  F.,  and  quench  in  oil.  The  temperature  need 
not  be  drawn  when  the  gears  are  quenched  in  oil.  The  final 
quenching  should  be  done  at  the  lowest  temperature  at  which 
the  piece  will  harden,  as  stated  in  the  first  method. 

A  small  gas  muffle  should  be  used  for  hardening.  A  properly 
constructed  gas  muffle  can  be  regulated  with  the  greatest  nicety, 
and  in  hardening  this  is  most  important.  When  steel  is  gradually 
heated,  there  is  a  certain  point  at  which  a  great  molecular  change 
takes  place,  and  perfect  hardness  can  be  obtained  only  by  quench- 
ing at  this  critical  point.  This  would  lie  between  1300  and  1450 
degrees  F.  When  steel  is  cooled,  whether  slowly  or  not,  it  bears 
in  its  structure  a  condition  representative  of  the  highest  tem- 
perature to  which  it  was  last  subjected.  From  this  it  will  be 
quite  clear  that  in  casehardening,  as  in  all  methods  of  hardening, 
the  steel  must  be  quenched  on  a  rising  heat. 

Steel  which  is  overheated  previous  to  the  final  quenching  is 
very  brittle  and  liable  to  fracture  easily,  and  although  quenched 
and  subsequently  hardened,  the  metal  has  little  or  no  cohesion, 
and  rapidly  wears  away.  Steel  so  hardened  breaks  with  a  very 
coarse  crystalline  fracture,  in  which  the  limits  of  the  case  are 
badly  defined.  If  quenching  takes  place  below  the  critical  point 
previously  mentioned,  the  steel  is  not  sufficiently  hard.  If 
above,  though  full  hardness  may  be  obtained,  strength  and 
tenacity  are  lost,  in  part  or  completely,  according  to  the  degree 
of  heat  by  which  the  critical  temperature  is  exceeded. 

It  may  be  asked  why  it  is  not  sufficient,  when  the  pieces  are 
heated  at  the  first  reheating  to  about  1500  degrees  F.,  to  place 
them  in  another  furnace  and  reduce  to  the  critical  tempera- 
ture and  quench,  instead  of  quenching  twice.  The  answer  is 
that  the  high  temperature  has  already  created  a  coarse  crystalline 
condition  in  the  steel,  and  that,  until  it  has  been  cooled  down 
below  the  critical  point,  and  reheated  to  the  critical  temperature, 
a  suitable  molecular  condition  cannot  be  obtained. 


MACHINING  AND  HEAT-TREATMENT 


179 


As  a  further  means  of  illustrating  what  is  meant  by  the  critical 
point,  Fig.  9  shows  a  curve  plotted  from  results  obtained  by  a 
recording  pyrometer,  in  which  the  decalescent  and  recalescent, 
or  critical  points  are  shown.  From  this  it  will  be  seen  that  the 


1500 


1400 


1300 


1200 
ul 


900 


800 


700 


DEC/ 


LESCENT  PO  NT 


SPECIMEN  REMOVED  FROM  FURNACE 


RECALESCENT  POINT 


\ 


4567 

TIME  IN  MINUTES 


10 

Machinery 


Fig.  9.    Curve  showing  the  Decalescent  and  Recalescent  Points 
for  Low-carbon  Nickel  Steel 

absorption  of  heat  occurred  at  about  1375  degrees  F.  on  the 
rising  temperature;  and  the  evolution  of  heat  at  1250  degrees  F. 
on  the  falling  temperature,  which  was  allowed  to  fall  slowly. 
The  relation  of  these  critical  points  to  hardening  is  that  it  cannot 


180  SPUR   GEARING 

take  place  unless  a  temperature  sufficient  to  produce  the  first 
action  is  reached  in  order  to  change  the  pearlite  carbon  to  harden- 
ing carbon;  and  also  unless  it  is  cooled  with  sufficient  rapidity 
to  eliminate  the  second  action.  The  temperature  would,  of 
course,  be  gaged  with  a  pyrometer. 

Sandblasting.  —  The  sandblasting,  which  serves  to  scour  off 
any  roughness  or  stains  which  have  been  left  on  the  surface 
during  the  heat- treatment,  etc.,  is  best  conducted  in  a  building 
separated  from  the  remainder  of  the  shop.  The  sand  should  be 
kept  in  a  bin  in  one  corner,  and  sucked  up  by  a  centrifugal  blower 
and  forced  by  air  pressure  through  a  blowpipe  which  terminates 
in  a  nozzle.  The  sand  being  forced  by  the  air  at  a  high  velocity 
may  be  directed  at  all  parts  of  the  piece  to  be  cleaned.  This  is 
one  of  the  most  efficient  methods  of  polishing  and  cleaning  the 
gears,  and  does  not  injure  the  hard  surface  in  any  way. 

Hardness  Testing.  —  The  gears  should  now  be  tested  for 
hardness.  The  scleroscope  appears  to  be  the  best  instrument 
for  doing  this.  If  a  gear  shows  a  considerable  drop  in  hardness, 
a  file  should  be  used  to  determine  whether  the  cause  is  due  to  the 
piece  not  being  hard,  or  to  crystallization.  If  the  part  can  be 
scratched  with  a  file,  it  shows  that  it  is  not  hard  enough.  If, 
however,  the  file  will  not  bite  on  the  spot  where  the  scleroscope 
reads  low,  then  it  is  positively  known  that  it  has  been  overheated 
and  is  crystallized.  A  good  method  of  testing  the  teeth  is  the 
drop  test.  By  this  method  a  ten-pound  weight  with  a  56-inch 
drop  is  directed  at  one  tooth.  The  number  of  blows  necessary 
to  break  the  tooth  should  be  noted.  This  test  would  only  be 
applied  occasionally,  say,  on  one  out  of  each  batch  of  gears.  A 
gear  which  has  given  satisfaction  should  be  tested,  and  the  result 
used  as  a  standard  for  future  comparison. 

Final  Turning  Operations  —  Grinding.  —  The  metal  left  for 
supporting  the  gears  while  undergoing  heat-treatment  is  now 
removed.  This  operation  should  be  carried  out  with  the  gear 
held  in  a  collet-chuck,  and  care  should  be  taken  that  the  outside 
diameter  runs  true  before  starting.  The  hole,  having  been  bored 
after  carbonizing,  is  now  soft.  The  metal  can  be  turned  out  on 
both  sides,  or  on  one  side,  as  the  case  may  be,  leaving  0.005  mcn 


MACHINING  AND   HEAT-TREATMENT  l8l 

on  the  face  of  the  web  for  grinding.  This  operation  should  come 
next,  and  is  most  essential  in  order  to  get  a  true  running  gear 
when  in  position.  The  castellated  gears  which  were  finish- 
bored  with  an  allowance  of  0.015  inch  before  carbonizing  are 
now  hard,  this  being  necessary  as  they  have  to  be  a  sliding  fit 
on  the  shaft.  The  part  of  the  bore  which  bears  against  the  lands 
of  the  castellated  shaft  is  all  that  can  be  ground.  Unfortunately 
no  method  has  yet  been  devised  for  grinding  the  castellations 
themselves,  and  these,  therefore,  have  to  be  lapped  separately 
by  hand  —  which  is  most  unsatisfactory.  While  grinding  the 
bore,  the  gear  should  be  held  in  a  collet-chuck  from  the  top 
diameter  —  hence  the  truing  of  the  top  diameter  with  the  pitch 
line  in  the  gear-cutting  process.  Gears  ground  in  this  way 
should  be  perfectly  true,  and  are  now  ready  to  be  tested  for  center 
distance  and  true  running. 

Drilling.  —  It  should  be  clearly  understood  that  the  web  of 
the  gear  shown  in  Fig.  i  is  now  soft.  This  brings  us  to  the  final 
machining  operation,  that  is,  the  drilling.  There  are  many  ad- 
vantages gained  in  leaving  this  operation  until  the  last.  The 
bore  is  now  the  correct  size,  this  being  essential  for  locating 
the  drilling  jig.  Furthermore,  in  the  grinding  operation,  the 
bore  is  totally  ignored,  the  top  diameter  being  the  most  im- 
portant. The  hole  has. been  ground  true  with  the  latter,  so 
that  if  the  holes  had  been  drilled  previous  to  the  hardening, 
they  would  not  be  concentric  with  the  bore,  and  would  not 
match  with  the  gear  or  center  to  which  it  has  to  be  bolted. 
Another  advantage  is  that  the  holes  are  now  soft  and  can  be 
reamed  with  the  piece  to  which  the  gear  is  to  be  bolted,  it  being 
understood  that  these  would  also  be  left  soft  in  the  parts  in  which 
the  holes  are  required.  All  these  are  important  points,  and  this 
is  the  reason  for  leaving  the  drilling  until  last. 

"  Running-in." — The  gears  should  now  be  bolted  to  their 
respective  parts  and  finally  "run  in"  before  being  placed  in  the 
car.  This  should  be  performed  in  a  special  case,  and  the  "run- 
ning-in"  done  under  belt  power.  The  bearings,  in  these  special 
cases,  are  set  at  the  proper  center  distance,  so  as  to  accommo- 
date the  various  gears  of  a  train,  thus  wearing  in  the  gears  so  that 


182  SPUR  GEARING 

all  those  for  similar  parts  are  absolutely  interchangeable.  The 
case  is  made  oil- tight,  and  a  mixture  of  finely  powdered  emery 
and  lubricating  oil  is  forced  through  an  opening  in  the  top,  so 
that  this  grinding  material  will  come  in  contact  with  all  the 
teeth  in  mesh  in  the  train.  The  grinding  is  continued  until  each 
tooth  has  been  worn  perfectly  smooth,  and  to  an  accurate  fit 
with  the  teeth  of  the  other  gears  with  which  it  comes  into  mesh. 
As  a  further  means  of  thoroughly  running  in  the  gears  of  the 
transmission  to  a  perfect  fit,  the  motor,  transmission  and  driving 
shaft  are  installed  in  the  chassis,  and  the  motor  is  run  while  the 
various  speeds  of  the  transmission  are  thrown  into  mesh.  During 
this  run  an  electric  dynamotor,  by  means  of  which  a  variable 
load  may  be  applied,  is  connected  to  the  end  of  the  driving  shaft. 
The  gears  should  now  be  as  perfect  as,  with  the  best  practice  yet 
known,  it  is  possible  to  get  them. 

It  may  be  objected  here  that  the  leaving  of  extra  metal  for 
heat-treatment  is  rather  costly.  This  is  a  question  upon  which 
there  is  room  for  considerable  difference  of  opinion.  It  seems 
that  if  the  method  of  leaving  extra  metal  is  used  with  discretion, 
and  only  in  the  case  of  very  intricate  gears,  the  most  satisfactory 
results  should  be  produced. 

Heat-treated  Gears  in  Machine  Tools.  —  The  requirements 
in  machine-tool  construction  differ  somewhat  from  those  in  the 
automobile  trade.  In  the  following  paragraphs  these  require- 
ments and  the  means  used  for  meeting  them  are  reviewed,  as 
stated  in  a  paper  read  before  the  National  Machine  Tool  Builders' 
Association  by  Mr.  A.  C.  Gleason.  The  methods  described  are 
those  in  use  in  the  Gleason  Works,  Rochester,  N.  Y.  The  points 
which  are  of  interest  to  the  average  machine  tool  builder  will  be 
considered.  The  methods  of  heat-treatment  described  here  are 
those  best  suited  to  the  requirements  at  the  Gleason  Works  and 
it  is  not  claimed  that  they  represent  general  practice.  In  fact, 
it  is  rather  difficult  to  say  what  is  the  general  practice,  as  the 
heat-treatment  of  steel  gears  is  largely  a  matter  of  individual 
requirements.  Methods  vary  considerably  even  among  manu- 
facturers of  a  certain  class  of  automobile  gears  which  are  of  a 
standard  shape  and  designed  for  one  purpose,  so  that  hardened 


MACHINING  AND  HEAT-TREATMENT  183 

gears  for  machine  tools  may  well  be  considered  a  distinct  propo- 
sition. Looking  at  this  subject  from  the  viewpoint  of  the  ma- 
chine tool  builder,  the  principal  points  to  be  considered  are: 

1.  The  advantages  to  be  gained  by  the  use  of  heat-treated 
gears. 

2.  The  selection  of  steel  to  suit  the  purpose  for  which  the 
gears  are  intended,  and  the  design  to  suit  hardening  conditions. 

3.  The  methods  of  hardening  and  the  necessary  equipment 
and  materials. 

4.  The  cost  of  heat-treated  gears. 

Advantages  of  Heat-treated  Gears.  —  The  advantages  in  the 
use  of  heat-treated  gears  properly  made  are  greatly  increased 
strength  and  hard  tooth  surfaces  which  resist  wear.  These 
points  are  certainly  of  vital  importance  in  modern  machine  tool 
design,  and  it  seems  inevitable  that  hardened  gears  will  very 
soon  be  in  general  use  in  this  class  of  machinery.  The  failure 
of  soft  metal  gears  to  stand  the  wear  and  tear  in  machine  tools 
is  too  frequently  the  cause  of  breakdowns,  and  in  spite  of  the 
rapid  development  in  design  in  other  ways  such  gears  are  still 
commonly  used.  They  are  a  serious  source  of  weakness  and  the 
logical  remedy  is  heat-treated  gears.  Heat-treated  gears  with 
their  increased  strength  and  ability  to  withstand  wear  offer 
almost  unlimited  opportunity  for  compact  design.  For  example, 
the  automobile  transmission  suggests  what  can  be  accomplished 
with  such  gears  in  making  machine  tools  more  convenient  in 
operation  and  more  durable. 

Steel  to  be  Used.  —  The  question  of  the  steel  to  be  used  de- 
pends not  only  upon  the  purpose  for  which  the  gears  are  in- 
tended but  also  upon  the  design.  When  the  gears  are  subjected 
to  severe  shock  or  heavy  overload  at  times,  a  steel  which  will 
show  the  greatest  tensile  strength  possible,  without  sacrificing 
toughness,  is  plainly  the  most  desirable;  but  steel  which  will 
show  these  qualities  has  certain  limitations  for  use  in  machine 
tool  construction,  and  it  will  be  interesting  to  note  the  value  of 
a  straight  hardening  steel  as  compared  with  the  more  commonly 
used  casehardening  steels.  Straight  hardening  steels  for  gears 
are  invariably  alloy  steels.  Chrome-nickel  alloy  steels  of  several 


184  SPUR  GEARING 

different  makes  show  an  increased  tensile  strength  of  150  per 
cent  after  heat-treatment.  The  analysis  varies  considerably  in 
different  makes,  requiring  a  corresponding  difference  in  the 
heat-treatment,  but  manufacturers  making  a  specialty  of  alloy 
steels  now  furnish  them  carefully  graded  with  instructions  for 
hardening  which  can  generally  be  relied  upon. 

There  are  frequent  exceptions,  however,  to  the  rules  for 
hardening  any  kind  of  gear  steel,  and  the  only  safe  method  is 
to  experiment  with  every  change  in  design.  It  is  not  sufficient 
to  cut  a  piece  off  the  bar  and  harden  it,  regardless  of  the  shape 
of  the  gear  being  made,  as  the  sample  piece  should  be  practically 
the  same  as  the  gear  in  order  to  produce  the  same  effect.  Take, 
for  example,  a  pinion  solid  on  a  shaft.  The  teeth  will  chill  much 
more  quickly  than  the  solid  section  under  them,  and  in  order  to 
avoid  shrinkage  strains  special  heat-treatment  is  required  to 
suit  this  shape  and  the  quality  of  steel  that  is  used.  If  a  sample 
pinion  of  this  size  were  made  separate  from  the  shaft,  the  effect 
of  the  same  heat-treatment  would  be  radically  different  owing 
to  the  fact  that  the  center  would  cool  almost  as  rapidly  as  the 
teeth. 

The  use  of  as  few  grades  of  steel  as  possible  is  advisable.  If 
the  steel  is  selected  to  suit  exactly  each  different  shape  of  gear 
and  the  purpose  for  which  it  is  intended,  it  would  necessitate  a 
large  variety  for  the  varied  requirements  of  the  machine  tool 
builder.  There  are  some  steels  which  are  more  adaptable  to 
different  conditions  in  hardening  than  others,  and  considering 
the  comparatively  small  quantities  and  the  variety  of  sizes  and 
shapes  in  gears  for  machine  tools,  it  is  well  to  keep  this  in  mind. 
" Fool-proof"  methods  in  the  heat- treatment  of  gears  are  far 
from  possible,  but  by  the  selection  of  a  few  grades  of  steel  which 
show  good  average  results,  the  work  can  be  greatly  simplified. 

Water  or  Oil-hardened  vs.  Casehardened  Gears.  —  On 
account  of  the  great  strength  and  toughness  of  chrome-nickel 
straight  hardening  steel  and  the  fact  that  it  hardens  clear  through, 
it  is  well  adapted  for  automobile  transmission  gears  and  any 
similar  purpose  in  the  construction  of  machine  tools.  Gears  of 
this  kind  do  not  chip  on  the  edges  of  the  teeth  and  will  stand  al- 


MACHINING  AND   HEAT-TREATMENT  185 

most  unlimited  hardship  in  sliding  in  and  out  of  mesh.  Chrome- 
nickel  straight  hardening  steels  tend  to  keep  their  shape  in 
hardening  better  than  the  low-carbon  casehardening  steels,  but 
they  do  warp  somewhat  and  they  cannot  be  straightened  without 
sacrificing  the  hardness  almost  entirely.  Gears  made  of  this 
steel  do  not  show  as  hard  a  bearing  on  the  surface  of  the  teeth 
as  those  made  of  casehardened  steel.  The  cost  of  machining 
alloy  straight  hardening  steels  will  average  at  least  twice  as  much 
as  that  of  casehardening  steels,  and  the  cost  of  the  steel  itself 
varies  from  fourteen  to  sixteen  cents  per  pound,  as  compared 
with  less  than  half  that  price  for  high-grade  alloy  casehardening 
steels. 

The  chief  advantage  in  the  use  of  casehardened  gears  is  the 
file-hard  tooth  surfaces.  The  easy  machining  qualities  and  low 
cost  of  the  stock  are  also  important  advantages,  but  the  superior 
wearing  qualities  of  casehardened  gears  make  them  the  best  at 
any  price  for  average  conditions  in  machine  tool  construction. 
In  designing  a  machine  tool  there  may  often  be  occasion  to  make 
the  gears  as  small  as  possible,  and  it  then  becomes  a  question 
as  to  the  choice  of  straight  hardening  steel  or  alloy  casehardening 
steel.  A  5  per  cent  nickel  low-carbon  casehardening  steel  will 
give  the  best  results  possible  if  strength  and  wearing  qualities 
are  considered,  although  if  strength  is  the  main  consideration, 
straight  hardening  steels,  as  previously  explained,  should  be  the 
choice. 

Of  the  alloy  casehardening  steels,  three  principal  grades  of 
nickel  steel  have  become  standard  to  a  large  degree  in  the  trade 
at  the  present  time.  These  are  5  per  cent  open-hearth  nickel 
alloy;  3!  per  cent  open-hearth  nickel;  and  i  to  i^  per  cent  nickel 
natural  alloy.  The  principal  characteristics  of  these  steels  are 
a  higher  tensile  strength  than  straight-carbon  steel,  and  a  corre- 
spondingly higher  strength  after  casehardening.  The  carbon 
case  has  a  close  bond  with  the  core  and  is  less  likely  to  chip  than 
the  ordinary  machinery  steel.  In  fact,  a  number  of  manufac- 
turers of  the  higher  grade  automobiles  use  the  5  per  cent  case- 
hardening  nickel  steel  in  their  change  speed  transmissions,  where 
straight  hardening  steels  might  be  regarded  as  more  favorable. 


1 86  SPUR  GEARING 

When  it  comes  to  substituting  casehardened  gears  for  soft  steel 
or  cast-iron  gears,  in  machine  tools,  it  will  generally  be  found 
that  a  straight-carbon  casehardening  steel  will  answer  every 
requirement,  and  unless  the  original  gears  have  been  consider- 
ably overloaded  the  more  expensive  alloy  steel  gears  would  be 
an  extravagance.  In  straight-carbon  steels  for  casehardening, 
0.15  to  0.25  per  cent  carbon  is  to  be  recommended.  A  lower 
degree  "of  carbon  than  this  is  likely  to  produce  a  laminated  case 
which  will  crack  or  chip  under  heavy  pressure.  Nickel  alloy 
steel  for  carbonizing  should  not  have  over  0.20  per  cent  carbon; 
o.io  to  0.20  per  cent  carbon  is  considered  the  best  for  this  pur- 
pose. The  nickel  alloy  has  practically  the  same  effect  in  harden- 
ing as  an  increase  in  carbon  of  about  o.io  per  cent.  A  0.25  per 
cent  carbon  nickel  alloy  steel,  casehardened,  will  generally 
harden  clear  through  very  much  the  same  as  a  straight  hardening 
steel. 

The  teeth  of  soft  steel  gears  in  machine  tools  seldom  break 
or  strip  until  they  have  worn  quite  thin,  and  long  experience 
goes  to  show  that  durability  is  generally  of  more  importance 
than  excessive  strength  in  gears  of  this  kind.  Casehardened 
steels  showman  increased  tensile  strength  ranging  from  30  per 
cent  in  standard  carbon  steel,  to  75  per  cent  in  high-grade  nickel 
alloy  steel.  The  straight-carbon  casehardening  steels  are  not 
as  uniform  in  analysis  as  the  alloy  steels  and,  therefore,  it  is  not 
always  possible  to  obtain  the  same  degree  of  hardness  in  the 
case  as  with  the  higher  grade  steels.  This  is  particularly  notice- 
able in  gears  of  heavy  section  hardened  in  oil,  but  it  rarely 
happens  that  the  hardness  will  drop  as  low  as  that  of  straight 
hardening  steels. 

Selective  or  Local  Hardening  of  Gears.  —  A  strong  point  in 
favor  of  casehardening  steels  for  machine  tool  gears  is  that 
certain  parts  can  readily  be  kept  soft  where  subsequent  fitting 
is  found  desirable.  Hub  projections,  web  surfaces,  etc.,  can  be 
copper-plated  or  enamelled  with  various  preparations  so  as  to 
exclude  the  carbon  in  the  carbonizing  heat,  leaving  these  parts 
soft  for  final  machining  operations.  Present-day  design  of  case- 
hardened  gears  for  machine  tools  often  makes  it  necessary  to 


MACHINING  AND   HEAT-TREATMENT 


l87 


keep  certain  parts  soft  in  order  to  avoid  shrinkage  strains  in  the 
hardening  process.  In  a  gear  with  light  hub  projections,  as 
shown  in  Fig.  10,  the  carbon  case  would  extend  almost  through 
the  thin  section  of  the  hub  at  the  keyseat,  making  it  quite  likely 
to  crack  on  account  of  the  uneven  shrinkage  in  hardening.  The 
making  of  gears  for  hardening  of  even  section  throughout  is 
advisable,  but  where  it  is  necessary  to  use  light  hub  projections 
like  the  one  shown,  they  can  be  made  safely  by  the  method  of 
selective  hardening. 


45  lys  p. 
9.m"o.D. 

MATE  25  TEETH 


Machinery 


Fig.    10.    Gear   with   Light    Hub 
kept  Soft  by  Copper  Plating 


Fig.  ii.    Gear  haying  Small  Pin- 
ion Integral  with  Hub 


In  selective  hardening,  copper-plating  the  surfaces  which  are 
to  be  kept  soft  is  the  most  satisfactory  method.  Generally  the 
parts  to  be  plated  can  be  immersed  in  the  solution,  leaving  the 
teeth  clear,  or  the  blank  can  be  copper-plated  all  over,  so  that 
when  the  teeth  are  cut,  the  bearing  surfaces  will  be  free  to  car- 
bonize. It  should  be  borne  in  mind  that  wherever  this  plated 
surface  is  marred,  carbon  will  enter  and  leave  a  hard  spot  after 
the  gears  are  hardened. 

Some  Points  in  Gear  Design.  —  Points  for  criticism  in  the 
design  of  the  pair  of  gears  shown  in  Fig.  10  are  the  extremely  long 
face  in  proportion  to  the  cone  distance  and  the  long  backing  or 


1 88  SPUR    GEARING 

overhang  of  the  pinion  on  its  bearing.  The  chances  are  all 
against  getting  a  full  bearing  of  the  teeth  throughout  the  length 
of  face,  after  the  gears  are  hardened  and  the  bores  are  ground  to 
size.  It  is  advisable  to  cut  down  the  length  of  face  to  not  more 
than  one- third  of  the  cone  distance  and  to  use  a  coarser  pitch 
with  a  smaller  number  of  teeth  for  the  same  diameters. 

Another  condition  which  it  is  necessary  to  guard  against  in 
gears  of  this  kind  occurs  where  a  small  pinion  is  made  solid  with 
the  gear  in  place  of  the  light  hub,  as  shown  in  Fig.  n.  There 
would  be  still  greater  risk  on  account  of  shrinkage  strains  in 
hardening  this  piece.  The  pinion  section  would  be  weak  when 
worked  out  of  the  center  of  bar  steel  and  it  would  be  far  better 
to  make  it  separate  from  the  gear. 

All  established  rules  for  the  horsepower  transmitted  by  gears 
are  based  on  the  use  of  soft  steel  or  cast  iron.  They  usually 
allow  a  stress  for  steel  of  two  and  one-half  times  that  of  cast  iron. 
This  may  be  correct  as  far  as  strength  is  concerned,  but  it  cer- 
tainly is  not  right  if  wear  is  to  be  taken  into  account.  Gears  of 
a  good  mixture  of  cast  iron,  showing  35  to  40  on  the  scleroscope 
test  for  hardness,  will  withstand  wear  fully  as  well  as  open- 
hearth  cast-steel  gears  of  the  same  size.  This  brings  up  the 
subject  of  wearing  qualities  of  casehardened  steel  gears  as  com- 
pared with  soft  gears.  A  number  of  electric  motor  drives  have 
been  equipped  with  casehardened  gears,  making  them  very 
much  smaller  than  the  soft  steel  gears  formerly  used,  with  most 
satisfactory  results,  and  judging  by  these  records,  a  stress  of  four 
times  the  usual  standard  allowed  for  cast  iron  in  standard  horse- 
power rules  such  as  the  Lewis  formula,  is  permissible. 

For  example,  in  a  30  horsepower  electric  drive  in  connection 
with  a  positive  pressure  blower,  the  original  soft  steel  gears, 
computed  according  to  the  standard  rule  for  horsepower,  re- 
quired a  pair  of  gears  having,  respectively,  49  and  16  teeth,  of 
3  diametral  pitch,  3|-inch  face;  the  casehardened  steel  gears 
which  have  been  in  use  now  for  several  years  have  49  and  16 
teeth,  of  4  diametral  pitch  and  2^-inch  face.  The  rule  making 
the  strength  of  casehardened  steel  gears  four  times  greater  than 
cast  iron  is  conservative,  as  the  gears  serving  as  a  basis  for  the 


MACHINING  AND   HEAT-TREATMENT  189 

calculations  are  considerably  larger  than  automobile  bevel 
driving  gears  and  transmit  more  power.  As  evidence  of  the 
advantage  of  casehardened  gears  over  hardened  alloy  steel  gears, 
may  be  cited  the  standard  automobile  practice  in  bevel  driving 
gears,  where  automobile  manufacturers  use  only  casehardened 
steels  for  such  bevel  driving  gears  at  the  present  time.  Their 
requirements,  as  is  well  known,  call  for  the  greatest  possible 
combined  durability  and  strength. 

The  quenching  of  casehardened  gears  at  the  lowest  heat 
possible  to  produce  the  required  degree  of  hardness  is  to  be 
recommended;  this  in  combination  with  slow  carbonizing  heats 
gives  excellent  results  with  one  quenching  heat.  In  the  use  of 
steels  suitable  for  casehardening,  experience  strongly  advises 
against  drawing  after  hardening.  Casehardening  steels  allow  of 
a  much  wider  range  in  the  heat-treatment  than  straight  harden- 
ing steels,  and  casehardened  gears  seem  decidedly  the  best  for 
general  purposes  in  machine  tools,  " clash  gears"  excepted. 

Equipment  for  Heat-treating  Gears.  —  The  Gleason  Works 
are  using  coal-burning  furnaces  for  carbonizing  in  preference  to 
oil  or  gas  (see  plan  of  heat-treating  department,  Fig.  12).  For 
continuous  use  with  slow  soaking  heats  coal  is  most  economical, 
and  with  very  little  attention  to  the  fires,  there  is  no  difficulty 
in  maintaining  and  controlling  the  heat.  The  fact  that  the 
coal-burning  furnaces  require  practically  no  attention  when  run 
after  working  hours,  and  that  no  power  is  required  for  air  pressure 
such  as  is  necessary  in  the  oil  or  gas  burners,  is  an  argument  in 
their  favor;  the  very  low  cost  of  fuel  is  also  an  important  con- 
sideration. Furnaces  of  this  kind,  however,  must  be  used  con- 
tinuously, day  after  day,  to  produce  the  best  results,  since  it 
requires  about  twenty-four  hours  from  the  start  to  bring  them 
up  to  a  carbonizing  heat.  On  the  other  hand,  there  is  no  delay 
whatever  in  getting  the  required  heat  with  oil  or  gas,  and  where 
there  is  not  sufficient  work  to  keep  a  furnace  running  continu- 
ously, the  oil  or  gas  furnace  would  undoubtedly  prove  the  most 
satisfactory.  An  oil  or  gas  furnace  is  necessary  in  any  case  for 
reheating,  so  that  the  same  furnace  can  often  be  used  to  advan- 
tage on  carbonizing  as  well.  Aside  from  the  cost  of  fuel,  the 


190 


SPUR  GEARING 


MACHINING  AND  HEAT-TREATMENT 


IQI 


objection  to  the  use  of  oil  or  gas  for  carbonizing  gears  is  that 
furnaces  of  this  kind  require  constant  attendance  when  run  after 
working  hours  and  that  extra  power  is  needed  for  air  pressure 
in  the  burners.  There  is  always  the  danger  of  an  interrupted 
flow  of  the  gas  or  air  to  be  guarded  against. 

In  regard  to  the  depth  of  carbon  case  in  gears,  ^  the  thickness 
of  teeth  at  the  pitch  line  is  recommended,  but  not  more  than  -^ 


3    PIPE  TO 
RESERVOIR 


FUEL  OIL  . 
STAND  PIPE 


V3TAND  PIPE 


Machinery 


Fig.  13.    Diagram  showing  Quenching  Oil  Reservoir  and 
Cooling  Apparatus 

inch  deep  in  the  coarsest  pitches  in  machine  tools.    According 
to  this  rule,  J-inch  pitch  should  have  3^ -inch  carbon  case. 

At  the  present  time,  there  are  several  makes  of  carbonizing 
compounds  extensively  used  which  have  proved  much  more 
satisfactory  than  bone  or  charred  leather.  Care  must  be  taken 
to  keep  these  compounds  perfectly  dry,  not  only  in  the  packing 
of  the  boxes  but  also  after  they  are  taken  out  to  cool;  if  any 
water  is  allowed  to  leak  in  when  the  material  is  hot,  a  chemical 
action  sets  up  which  has  the  effect  of  blistering  the  casehardened 
surfaces  of  the  gears,  the  same  as  if  they  were  overheated.  Short 


SPUR   GEARING 


pieces  of  common  machinery  steel  about  \  inch  square  are 
generally  placed  in  the  top  of  the  boxes  for  test  pieces,  and 
before  the  work  is  taken  out  of  the  box  these  pieces  should  be 
hardened  and  broken  to  make  sure  that  the  depth  of  case  is 
right. 

It  is  a  well-known  fact  that  nickel  alloy  steels  require  a  longer 
carbonizing  heat  than  straight-carbon  steel,  and  in  order  to 
determine  the  proper  depth  of  case,  test  pieces  of  the  same 
material  should  be  used  with  the  plain  carbon  steel  test  pieces  so 


Machinery 


Fig.  14.    Oil  or  Water  Quenching  Tank 

as  to  make  a  proper  comparison.  It  does  not  pay  to  use  plain 
cast-iron  carbonizing  boxes;  semi-steel  is  better  and  cast  steel  is 
the  best. 

Oil  used  for  Quenching.  —  The  Gleason  Works  use  mineral 
oil  for  quenching  which  has  a  3io-degree  flash  test  and  viscosity 
of  74  inches  at  104  degrees.  It  is  a  thin  petroleum  oil  which  can 
be  bought  for  17  cents  per  gallon  by  the  barrel.  With  tempering 
oils  like  this  a  safe  rule  is  to  quench  one  pound  of  steel  to  a  gallon, 
every  four  hours,  where  no  special  arrangements  are  made  for 
cooling  the  oil.  With  a  cooling  system,  as  shown  in  Fig.  13,  it 


MACHINING  AND  HEAT-TREATMENT  193 

is  possible  to  quench  3000  pounds  of  steel  in  eight  hours  with  700 
gallons  of  oil.  The  tempering  oil,  as  shown,  is  circulated  through 
the  inside  of  a  radiator,  and  the  fuel  oil  is  circulated  through  the 
outer  jacket.  The  radiator  is  simply  a  powerhouse  water  heater 
which  is  adapted  for  this  purpose  without  change.  The  fuel  oil 
is  used  for  cooling  because  of  its  convenience.  Greater  efficiency, 
of  course,  could  be  obtained  by  having  a  flow  of  water  for  the 
purpose  or  by  increasing  the  radiating  surface.  The  radiator 
has  thirty  feet  of  cooling  surface.  There  has  never  been  any 
difficulty  with  overheating  of  the  oil  with  this  system,  no  matter 
how  fast  the  work  is  put  through;  occasionally  the  temperature 
is  as  high  as  120  degrees  but  never  any  higher.  Quenching  can 
be  done  about  twice  as  fast  as  without  the  cooling  system.  Fig. 
14  shows  the  form  of  oil  or  water  quenching  tank  used. 

Cost  of  Heat-treated  Gears.  —  In  leading  up  to  the  final  cost 
of  heat-treated  gears,  the  actual  cost  of  labor  and  materials  in 
the  heat-treatment  of  chrome-nickel  straight  hardening  steels  as 
compared  with  casehardening  in  general  is  here  presented. 

Cost  data  for  1000  pounds  of  gears  made  of  chrome-nickel 
straight  hardening  steel: 

Labor  (one  working  foreman  and  two  assistants,  wages)  $  9.85 
Fuel  oil  (two  hardening  furnaces,  60  gallons)      ....       3  .  oo 

Quenching  oil,  1  1  gallon     .............       0.25 

Tempering  oil,  2  gallons     ..........    ...       o  .  50 

Pyrometer  ends    .................       o.io 

Gas  for  drawing  temper,  1500  cu.  ft  ..........       1.45 

$15.15  -5-  looo  =  $0.0152  per  pound  *J5  •  J5 

Cost  data  for  1000  pounds  of  casehardened  gears: 
Labor  (one  working  foreman  and  two  assistants,  wages)    $9.85 

Coal  (3  furnaces,  200  pounds  each)     .........  2  .  oo 

Carbonizing  compound      .    .    ............  i  .  oo 

Fuel  oil  (two  hardening  furnaces,  60  gallons)    .    .    .    ...      3.00 

Quenching  oil,  ij  gallon    ..............  o.  25 

Pyrometer  ends  ...............    ...  0.25 

Carbonizing  boxes  (average)     ......    ......  2.50 

Wear  and  tear  on  3  carbonizing  furnaces  .......  0.30 


$19.15  -5-  1000  =  $0.0192  per  pound  *J9-  X5 


194  SPUR  GEARING 

No  account  is  made  of  the  cost  of  power  or  other  overhead 
expenses  as  they  are  the  same  in  either  case. 

Following  is  a  tabulated  account  of  the  actual  cost  of  labor 
and  materials  in  making  up  a  small  lot  of  miter  gears  complete 
from  bar  stock  using  the  various  grades  of  steel  referred  to : 

Taking  a  miter  gear  having  18  teeth  of  4  pitch,  ij  inch  face, 
1 1  inch  bore  and  an  ordinary  hub,  the  weight  of  the  rough  bar 
stock  is  1 1  pounds  and  the  finished  gear,  5 \  pounds.  The  labor 
cost  for  the  machine  work  would  be  practically  the  same  for  this 
gear,  in  any  of  the  standard  steels  for  carbonizing.  The  only 
difference  in  the  complete  cost  would  be  in  the  stock.  According 
to  this,  and  taking  the  labor  cost  at  $i,  the  cost  of  the  gear  com- 
plete would  be  as  follows: 

Straight  carbon  steel  at  3  cents .  .    .    .  $i .  44 

i  to  ij  per  cent  natural  alloy  steel  at  4^  cents 1.61 

35  per  cent  O.  H.  nickel  steel  at  6  cents  .    . .  .    .   .    .    .    .  1.77 

5  per  cent  O.  H.  nickel  steel  at  8  cents    ....   .   ...  i .  99 

The  cost  of  machine  work  is  practically  the  same  in  either  case- 
hardening  straight-carbon  steel  or  any  of  the  nickel  alloy  case- 
hardening  stock.  Heat-treated  gears  in  machine  tools  are  on  the 
side  of  superior  quality  and  greatest  efficiency.  It  is  safe  to  say 
that  within  the  next  few  years,  soft  steel  gears  in  machine  tools 
will  become  a  thing  of  the  past,  just  as  gears  with  cast  teeth 
were  abandoned  twenty  years  ago. 

Heat-treatment  Methods  used  at  the  Boston  Gear  Works.  — 
As  all  the  work  usually  carried  on  in  an  up-to-date  hardening 
plant  is  handled  in  connection  with  the  manufacture  of  gears  at 
the  Boston  Gear  Works,  a  brief  review  of  the  methods  here  in 
vogue  will  be  of  value.  In  this  plant  the  various  heat-treating 
processes  carried  out  consist  of  carbonizing  and  hardening  low- 
carbon  and  alloy  steels,  oil  hardening  high-carbon  and  alloy 
steels,  heat-treating  high-carbon  and  tungsten  tool  steels,  pack- 
hardening,  annealing  and  drawing  the  temper,  all  the  work 
consisting  mainly  of  gears. 

The  heat-treatments  in  use  are  mainly  those  recommended  by 
the  Iron  and  Steel  Division  of  the  Society  of  Automobile  Engi- 
neers, known  as  the  "S.  A.  E."  treatments.  These  treatments 


MACHINING  AND  HEAT-TREATMENT  195 

cover  carbon  steels  with  o.io  to  0.95  per  cent  carbon;  carbon 
steel  screw  stock;  3^  per  cent  nickel  steel  with  0.15  to  0.50  per 
cent  carbon;  low,  medium  and  high  nickel-chromium  steel  with 
0.15  to  0.50  per  cent  carbon;  low  and  medium  nickel-chrome- 
vanadium  steel;  i.o  and  1.20  per  cent  chromium  steels  with  0.95 
and  i. 20  per  cent  carbon;  chrome- vanadium  steel  with  0.15  to 
0.95  per  cent  carbon;  and  silico-manganese  steel  with  0.50  per 
cent  carbon. 

Bands  of  Heat-treatments  Used.  —  For  special  gear  steels, 
referred  to  in  the  following  list,  special  treatments  are  used  which 
have  been  largely  developed  in  the  plant. 

For  0.08  to  o.  15  per  cent  carbon  No.  i  machinery  steel.  Treat- 
ment Z. 

For  0.15  to  0.25  per  cent  carbon  No.  2  machinery  steel.  Treat- 
ment 7. 

For  0.35  to  0.40  per  cent  carbon  No.  3  machinery  steel.  Treat- 
ment E. 

For  0.40  to  0.50  per  cent  carbon  oil  hardening  No.  4  machinery 
steel.  Treatment  /. 

For  0.15  to  0.25  per  cent  carbon  3^  per  cent  nickel  steel. 
Treatment  G. 

For  0.15   to  0.25  per  cent  carbon  chrome- vanadium  steel. 

Treatment  S. 

Treatment  G 

1.  Carbonize  between  1600  and  1750  degrees  F.  (1650-1700 
degrees  F.  desired). 

2.  Cool  slowly  in  the  carbonizing  material. 

3.  Reheat  to  from  1450  to  1525  degrees  F. 

4.  Quench. 

5.  Reheat  to  from  1300  to  1400  degrees  F. 

6.  Quench. 

7.  Reheat  to  from  250  to  500  degrees  F.  (in  accordance  with 
the  necessities  of  the  case),  and  cool  slowly. 

Treatment  S 

1.  Carbonize  at  a  temperature  between  1600  and  1750  degrees 
F.  (1650-1700  degrees  F.  desired). 

2.  Cool  slowly  in  the  carbonizing  mixture. 


196  SPUR   GEARING 

3.  Reheat  to  from  1600  to  1700  degrees  F. 

4.  Quench. 

5.  Reheat  to  from  1475  to  1550  degrees  F. 

6.  Quench. 

7.  Reheat  to  from  250  to  550  degrees  F.,  and  cool  slowly. 

Treatment  E 

1.  Heat  to  from  1500  to  1550  degrees  F. 

2.  Cool  slowly. 

3.  Reheat  to  from  1400  to  1450  degrees  F. 

4.  Quench. 

5.  Reheat  to  from  600  to  1200  degrees  F.  and  cool  slowly. 

Treatment  Z 

1.  Carbonize  at  1600  to  1650  degrees  F. 

2.  Quench  direct  in  oil. 

3.  Draw  in  oil  at  from  400  to  450  degrees  F. 

Treatment  I 

1.  Carbonize  at  1600  to  1650  degrees  F. 

2.  Cool  in  pot. 

3.  Reheat  from  1550  to  1600  degrees  F.  (1575  degrees  pre- 
ferred). 

4.  Quench  in  oil. 

5.  Reheat  from  1350  to  1400  degrees  F.  (1375  degrees  pre- 
ferred). 

6.  Quench  in  oil. 

7.  Draw  at  from  300  to  400  degrees  F. 

Treatment  J 

1.  Heat  to  from  1450  to  1500  degrees  F.  (1480  degrees  F. 
preferred). 

2.  Quench  in  oil. 

3.  Reheat  to  from  Boo  to  900  degrees  F.     (Sometimes  drawn 
as  low  as  600  degrees  F.  or  as  high  as  noo  degrees  F.,  according 
to  requirements.) 

4.  Cool  slowly  in  the  air. 

There  is  considerable  doubt  relative  to  the  proper  heat-treat- 
ment for  0.35  to  0.40  per  cent  carbon  machinery  steel,  as  it  is 


MACHINING  AND  HEAT-TREATMENT  197 

neither  a  casehardening  nor  oil  hardening  steel.  This  steel  is 
largely  used  in  its  natural  state.  Treatment  E  is  the  one  recom- 
mended by  the  S.  A.  E.  for  the  treatment  of  0.40  per  cent  carbon 
steel.  Treatments  G  and  6"  are  also  standard  treatments  of  the 
S.  A.  E. 

Tests  on  Heat-treated  Gears.  —  The  method  used  by  the 
Boston  Gear  Works  for  testing  the  heat- treatment  of  steel  is  to 
cut  off  J-inch  disks  from  both  ends  of  a  bar  after  the  rough  ends 
have  been  cut  off.  The  bar  is  stamped  with  a  number  and  the 
disks  with  the  same  number.  The  disks  are  then  heat-treated 
and  broken  by  a  ten-pound  drop  hammer,  the  height  of  drop  and 
number  of  blows  being  recorded.  Gear  teeth  are  also  tested  by 
a  Burgess  drop  hammer  testing  machine,  the  standard  being  a 
i5-tooth,  6-pitch,  i4|-degree  involute  gear  of  one-inch  face.  The 
average  of  a  number  of  tests  on  casehardened  standard  test 
gears  of  No.  2  machinery  steel  required  31  blows  with  a  ten- 
pound  hammer  dropped  thirty  inches  to  fracture  a  tooth. 

Carbonizing  Methods  of  the  Boston  Gear  Works.  —  The 
depth  and  quality  of  the  case  that  is  produced  depends  upon  the 
duration  of  the  heat-treatment,  the  temperature,  the  kind  of 
carbonizing  material  used,  the  cooling  process,  and  the  tempera- 
ture at  which  reheating  and  quenching  operations  are  carried  on. 
For  casehardened  gears,  the  best  results  are  obtained  with  a 
refined  case  and  fibrous  core.  Great  care  is  required  in  packing 
gears  in  the  pots  with  the  carbonizing  material.  The  pot  covers 
are  luted  or  sealed  with  fireclay  and  it  is  important  that  no  gas 
escapes.  More  heats  are  obtained  with  malleable  iron  pots  than 
with  those  of  ordinary  cast  iron,  as  the  latter  burn  out  more 
rapidly.  Granulated  bone,  charcoal  and  commercial  carbonizing 
compounds  are  used  for  carbonizing. 

The  casehardening  pots  for  carbonizing  automobile  main- 
drive  ring  bevel  gears  have  cored  centers  and  lugs  on  the  bot- 
tom to  allow  a  more  uniform  circulation  of  heat  than  is  secured 
with  the  use  of  ordinary  cylindrical  pots.  The  carbonizing 
material,  in  this  case  a  compound,  is  packed  around  the  ring 
gears  in  the  pots,  after  which  the  covers  are  sealed  on  with 
fireclay. 


198 


SPUR   GEARING 


Casehardening.  —  After  carbonizing,  the  work  is  carefully 
reheated  and  treated  according  to  one  of  the  preceding  methods. 
The  quenching  equipment  includes  a  cylindrical  steel  water 
cooling-tank,  three  feet  in  diameter  by  six  feet  deep,  with  a  water 
inlet  and  outlet.  The  tank  is  placed  in  a  vertical  position  with 
half  of  its  depth  below  the  floor  level.  A  similar  oil  cooling-tank, 
thirty  inches  in  diameter  by  six  feet  deep,  is  placed  in  an  outer 
water  tank,  three  feet  in  diameter  by  six  feet  deep,  which  acts  as 
a  water  jacket.  This  outer  tank  has  a  water  inlet  and  outlet 
which  provides  circulation  that  keeps  the  oil  at  a  low  temperature. 


Machinery 


Fig.  15.    Device  for  Testing  Gears 

Device  for  Testing  and  Measuring  Gears.  —  Various  devices 
have  been  designed  for  testing  the  accuracy  of  the  diameters  and 
tooth  outlines  of  gears.  A  very  simple  testing  device  consists 
of  a  cast-iron  plate  in  which  are  accurately  located  holes  for 
studs  on  which  the  gear  to  be  tested  can  be  placed  so  that  its 
running  with  a  master  gear  can  be  inspected.  A  more  elaborate 
device  for  the  testing  of  gears  is  shown  in  Fig.  15.  This  device 
has  been  used  for  several  years  with  very  satisfactory  results. 
Differences  in  diameter  and  eccentricity  of  0.0004  mch  can  De 
measured  by  it  and  the  exact  meshing  of  a  couple  of  spur  gears 
can  be  accurately  tested.  The  illustration  shows  a  longitudinal 
section  of  the  device  on  a  cast-iron  base  with  fitted  tool  slides, 
each  provided  with  a  spindle  to  hold  the  gears  to  be  tested.  The 


MACHINING  AND   HEAT-TREATMENT  199 

slide  to  the  right  is  moved  by  means  of  a  long  adjusting  screw. 
The  one  to  the  left  has  only  limited  adjustment  but  transfers 
its  motion  greatly  magnified  to  a  pointer  placed  on  the  front  side 
of  the  base.  The  illustration  clearly  shows  how  the  motion  is 
transferred  to  the  index  which  measures  the  differences  in  eccen- 
tricity and  errors  in  the  meshing  of  the  gears.  To  detect  dif- 
ferences in  eccentricity,  it  is  most  convenient  to  place  the  gear 
to  be  tested  on  one  spindle  and  to  use  a  blank  with  a  single  tooth 
on  the  other.  This  tooth  is  meshed  in  succession  with  all  the 
teeth  of  the  gear  being  tested  and  by  observing  the  different 
positions  of  the  pointer  while  each  tooth  is  measured,  the  eccen- 
tricity may  be  determined  with  great  accuracy. 


CHAPTER  X 


BEVEL   GEAR   RULES   AND   FORMULAS 

Bevel  Gear  Definitions. — Bevel  gearing  is  used  for  transmitting 
motion  between  shafts  the  center  lines  of  which  intersect.  The 
teeth  of  bevel  gears  are  constructed  on  imaginary  pitch  cones  in 
the  same  way  that  the  teeth  of  spur  gears  are  constructed  on 
imaginary  pitch  cylinders.  In  Fig.  i  is  shown  a  drawing  of  a 
pair  of  bevel  gears  of  which  the  gear  has  twice  as  many  teeth  as 
the  pinion.  The  latter  thus  revolves  twice  for  every  revolution 


PITCH  CONE  ANGLE  = 


Machinery 


Fig.  i.    Bevel  Gear  and 
Pinion 


Fig.  2.    Pitch  Cones  of  Gears  in 
Fig.  i 


of  the  gear.  In  Fig.  2  is  shown  (diagrammatically)  a  pair  of 
conical  pitch  surfaces  driving  each  other  by  frictional  contact. 
The  shafts  are  set  at  the  same  center  angle  with  each  other,  as 
in  Fig.  i,  and  the  base  diameter  of  the  gear  cone  is  twice  that 
of  the  pinion  cone,  so  that  the  latter  will  revolve  twice  to  each 
revolution  of  the  former.  This  being  the  case,  the  cones  shown 
in  Fig.  2  are  the  pitch  cones  of  the  gears  shown  in  Fig.  i.  The 
term  "pitch  cone"  may  be  defined  as  follows:  The  pitch  cones 
of  a  pair  of  bevel  gears  are  those  cones  which,  when  mounted 
on  the  shafts  in  place  of  the  bevel  gears,  will  drive  each  other  by 


200 


RULES  AND   FORMULAS 


201 


frictional  contact  in  the  same  velocity  ratio  as  given  by  the  bevel 
gears  themselves. 

The  pitch  cones  are  determined  by  their  pitch  cone  angles,  as 
shown  in  Fig.  2.  The  sum  of  the  two  pitch  cone  angles  equals 
the  center  angle,  the  latter  being  the  angle  made  by  the  shafts 


Machinery 


Fig.  3.  Right-angle  Bevel  Gearing          Fig.  4.    Miter  Gearing 

with  each  other,  measured  on  the  side  on  which  the  contact 
between  the  cones  takes  place.  The  center  angle  and  the  pitch 
cone  angles  of  the  gear  and  the  pinion  are  indicated  in  Fig.  i. 

Different  Bands  of  Bevel  Gears.  —  In  Fig.  3  is  shown  a  pair  of 
bevel  gears  in  which  the  center  angle  (7)  equals  90  degrees,  or  in 


Machinery 


Fig.  5.   Acute-angle  Bevel  Gearing       Fig.  6.  Obtuse-angle  Bevel  Gearing 

other  words,  the  figure  shows  a  case  of  right-angle  bevel  gearing. 
To  the  special  case  shown  in  Fig.  4  in  which  the  number  of  teeth 
in  the  two  gears  is  the  same,  the  term  miter  gearing  is  applied; 
here  the  pitch  cone  angle  of  each  gear  will  always  equal  45 
degrees. 
When  the  center  angle  is  less  than  90  degrees  we  have  acute- 


202 


BEVEL   GEARING 


angle  bevel  gearing,  as  shown  in  Fig.  5.  When  the  center  angle 
is  greater  than  90  degrees,  we  have  obtuse-angle  bevel  gearing, 
shown  in  Fig.  6  and  also  in  Fig.  i .  Obtuse-angle  bevel  gearing  is 
met  with  occasionally  in  the  two  special  forms  shown  in  Figs.  7 
and  8.  When  the  pitch  cone  angle  ag  equals  90  degrees,  the  gear 
is  called  a  crown  gear.  In  this  case  the  pitch  cone  evidently 
becomes  a  pitch  plane,  or  disk.  When  the  pitch  cone  angle  of 
the  gear  is  more  than  90  degrees,  as  in  Fig.  8,  this  member  is 
called  an  internal  bevel  gear,  and  its  pitch  cone  when  drawn  as 
in  Fig.  2,  would  mesh  with  the  pitch  cone  of  the  pinion  on  its 
internal  conical  surface.  These  two  special  forms  of  gears  are 
of  rare  occurrence. 


Machinery 


Fig.  7.    Crown  Gear  and  Pinion 


Fig.  8.    Internal  Bevel  Gear  and 
Pinion 


Bevel  Gear  Dimensions  and  Definitions.  —  In  Fig.  9,  which 
shows  an  axial  section  of  a  bevel  gear,  the  pitch  lines  show  the 
location  of  the  periphery  of  the  imaginary  pitch  cone.  The 
pitch  cone  angle  is  the  angle  which  the  pitch  line  makes  with  the 
axis  of  the  gear.  The  pitch  diameter  is  measured  across  the  gear 
drawing  at  the  point  where  the  pitch  lines  intersect  the  outer  edge 
of  the  teeth.  The  teeth  of  bevel  gears  grow  smaller  as  they  ap- 
proach the  vertex  of  the  pitch  cone,  where  they  would  disappear 
if  the  teeth  were  cut  for  the  full  length  of  the  face.  In  speaking 
of  the  pitch  of  a  bevel  gear  we  always  mean  the  pitch  of  the  larger 
or  outer  ends  of  the  teeth.  Diametral  and  circular  pitch  have 
the  same  meaning  as  in  the  case  of  spur  gears,  the  diametral  pitch 


RULES   AND   FORMULAS 


203 


T" 


If 


1 


•*.  .©• 

-C^>v  3    ? 

xs*»._*te_j  3    » 
V\1I   ! 


a  a> 

opq 

' 


|j 

o> 


204  BEVEL  GEARING 

being  the  number  of  teeth  per  inch  of  the  pitch  diameter,  while 
the  circular  pitch  is  the  distance  from  the  center  of  one  tooth  to 
the  center  of  the  next,  measured  along  the  pitch  diameter  at  the 
back  faces  of  the  teeth.  The  addendum  is  the  height  of  the  tooth 
above  the  pitch  line  at  the  large  end.  The  dedendum  (the  depth 
of  the  tooth  space  below  the  pitch  line)  and  the  whole  depth  of 
the  tooth  are  also  measured  at  the  large  end. 

The  pitch  cone  radius  is  the  distance  measured  on  the  pitch  line 
from  the  vertex  of  the  pitch  cone  to  the  outer  edge  of  the  teeth. 
The  width  of  the  face  of  the  teeth,  as  shown  in  Fig.  9,  is  measured 
on  a  line  parallel  to  the  pitch  line.  The  addendum,  whole  depth 
and  thickness  of  the  teeth  at  the  small  or  inner  end,  may  be 
derived  from  the  corresponding  dimensions  at  the  outer  end,  by 
calculations  depending  on  the  ratio  of  the  width  of  face  to  the 
pitch  cone  radius.  (See  s,  w  and  t  in  Fig.  12.) 

The  addendum  angle  is  the  angle  between  the  top  of  the  tooth 
and  the  pitch  Ijne.  The  dedendum  angle  is  the  angle  between 
the  bottom  of  the  tooth  space  and  the  pitch  line.  The  face  angle 
is  the  angle  between  the  top  of  the  tooth  and  a  perpendicular  to 
the  axis  of  the  gear.  The  edge  angle  (which  equals  the  pitch 
cone  angle)  is  the  angle  between  the  outer  edge  and  the  per- 
pendicular to  the  axis  of  the  gear.  The  latter  two  angles  are 
measured  from  the  perpendicular  instead  of  from  the  axis,  for 
the  convenience  of  the  workman  in  making  measurements  with 
the  protractor  when  turning  the  blanks.  The  cutting  angle  is  the 
angle  between  the  bottom  of  the  tooth  space  and  the  axis  of  the 
gear. 

The  angular  addendum  is  the  height  of  tooth  at  the  large  end 
above  the  pitch  diameter,  measured  in  a  direction  perpendicular 
to  the  axis  of  the  gear.  The  outside  diameter  is  measured  over 
the  corners  of  the  teeth  at  the  large  end.  The  vertex  distance 
is  the  distance  measured  in  the  direction  of  the  axis  of  the  gear 
from  the  corner  of  the  teeth  at  the  large  end  to  the  vertex  of  the 
pitch  cone.  The  vertex  distance  at  the  small  end  of  the  tooth 
is  similarly  measured. 

The  shape  of  the  teeth  of  a  bevel  gear  may  be  considered  as 
being  the  same  as  for  teeth  in  a  spur  gear  of  the  same  pitch  and 


RULES   AND   FORMULAS 


205 


style  of  tooth,  having  a  radius  equal  to  the  distance  from  the 
pitch  line  at  the  back  edge  of  the  tooth  to  the  axis  of  the  gear, 
measured  in  a  direction  perpendicular  to  the  pitch  line.  This 


Df  . 
distance  is   dimensioned  —  in  Fig.   12. 

2 


The  number  of  teeth 


which  such  a  spur  gear  would  have,  as  determined  by  diameter  D' 
thus  obtained,  may  be  called  the  "  number  of  teeth  in  equivalent 
spur  gear,"  and  is  used  in  selecting  the  cutter  for  forming  the 
teeth  of  bevel  gears  by  the  formed  cutter  process. 


Machinery 


Fig.  12.    Diagram  used  in  Connection  with  Derivation  of  Formulas 
for  Bevel  Gears 

In  two  special  forms  of  gears,  the  crown  gear,  Fig.  10,  and  the 
internal  bevel  gear,  Fig.  n,  the  same  dimensions  and  definitions 
apply  as  in  regular  bevel  gears,  though  in  a  modified  form  in 
some  cases.  In  the  crown  gear,  for  instance,  the  pitch  diameter 
and  the  outside  diameter  are  the  same,  and  the  pitch  cone 
radius  is  equal  to  |  the  pitch  diameter.  The  addendum  angle 
and  the  face  angle  are  also  the  same.  The  angular  addendum 
becomes  zero,  and  the  vertex  distance  is  equal  to  the  adden- 


206  BEVEL  GEARING 

dum.  The  number  of  teeth  in  the  equivalent  spur  gear  becomes 
infinite,  or  in  other  words,  the  teeth  are  shaped  like  those  of  a 
rack. 

When  the  pitch  cone  angle  is  greater  than  90  degrees,  so  that 
the  gear  becomes  an  internal  bevel  gear,  as  in  Fig.  n,  the  out- 
side diameter  (or  edge  diameter  as  it  is  better  called  in  the  case 
of  internal  gears)  becomes  less  than  the  pitch  diameter.  Other- 
wise the  conditions  are  the  same  although  many  of  the  dimensions 
are  reversed  in  direction. 

Rules  and  Formulas.  —  Rules  and  formulas  for  calculating 
the  dimensions  of  bevel  gears  are  given  on  the  following  pages. 
In  these  formulas  the  reference  letters  below  are  used: 

N  =  number  of  teeth; 
P  =  diametral  pitch; 
Pf  =  circular  pitch; 

7T    =    3.I4I6; 

a  =  pitch  cone  angle  and  edge  angle; 

7  =  center  angle; 
D  =  pitch  diameter; 
S  =  addendum; 

S  +  A  =  dedendum  (A  =  clearance) ; 
W  =  whole  depth  of  tooth  space; 
T  =  thickness  of  tooth  at  pitch  line; 
C  =  pitch  cone  radius; 
F  =  width  of  face; 

s  =  addendum  at  small  end  of  tooth; 

/  =  thickness  of  tooth  at  pitch  line  at  small  end; 

0  =  addendum  angle; 

<£  =  dedendum  angle; 

8  =  face  angle; 

f  =  cutting  angle; 
K  =  angular  addendum; 

O  =  outside  diameter  (edge  diameter  for  internal  gears) ; 
J  =  vertex  distance; 
j  =  vertex  distance  at  small  end; 
N'  =  number  of  teeth  in  equivalent  spur  gear. 


RULES  AND   FORMULAS  207 

Subp  refers  to  dimensions  applying  to  pinion  (ap,  Np,  etc.). 


refers  to  dimensions  applying  to  gear  (aoj  Ng,  etc.). 

It  will  be  noted  that  directions  for  the  use  of  these  rules  are 
given  for  each  of  the  six  cases  of  right-angle  bevel  gearing,  miter 
bevel  gearing,  acute-angle  and  obtuse-angle  bevel  gearing,  and 
crown  and  internal  bevel  gears.  Examples  are  given  on  sub- 
sequent pages  which  show,  in  detail,  the  use  of  the  rules  and 
formulas. 

Derivation  of  Formulas  for  Bevel  Gear  Calculations.  —  The 
derivation  of  most  of  these  formulas  is  evident  on  inspection  of 
Figs,  i  to  12  inclusive,  for  anyone  who  has  a  knowledge  of  ele- 
mentary trigonometry.  It  is  not  necessary  to  know  how  they 


Machinery 


Fig.  13.    Diagram  for  Obtaining  Pitch  Cone  Angle  of  Acute-angle 
Bevel  Gearing 

were  derived  in  order  to  use  them,  as  all  that  is  needed  is  the 
ability  to  read  a  table  of  sines  and  tangents. 

Formulas  (5),  (6),  (7)  and  (8)  are  the  same  as  for  Brown  & 
Sharpe  standard  gears.  The  dimensions  at  the  small  end  of  the 
tooth  given  by  Formulas  (10),  (n)  and  (19)  obviously  are  to  the 
corresponding  dimensions  at  the  large  end,  as  the  distance  from 
the  small  end  of  the  tooth  to  the  vertex  of  the  pitch  cone  is  to  the 
pitch  cone  radius.  This  relation  is  expressed  by  these  formulas. 
The  derivation  of  Formula  (20)  may  be  understood  by  reference 
to  Fig.  1  2  : 


; 

cos  a      PXcosa  P 

Nr  N  A7,        N 

therefore       —  =  rr—  -  ,     or     Nf  = 


P       PXcosa'  cosa 


208 


BEVEL  GEARING 


Formula  (21)  for  checking  the  calculations  will  also  be  under- 
stood from  Fig.  12,  where  it  will  be  seen  that 

r 
O  =  2  ab  X  cos  5,  and  that  ab  = 


COS0 


Therefore, 


0 


2  C  X  COS  6 

cos  e 


Formulas  (22)  to  (27)  inclusive  are  simply  the  corresponding 
Formulas  (i),  (9),  (14),  (15),  (16)  and  (20)  when  a  —  45  degrees. 
Formula  (28)  is  derived  as  shown  in  Fig.  13 : 

e 


c  = 


e  i  .   j         d 

,     also,     c  =  a  +  b  =  ~ 


tan  a 


sin  7      tan  7 


Machinery 


Fig.  14.    Diagram  for  Obtaining  Pitch  Cone  Angle  of  Obtuse-angle 
Bevel  Gearing 


Therefore. 


-  •  =  -  --  1  -- 
tan  ap      sin  7      tan  7 


^  (sin  7  X  tan  7) 
7^^  -  -  -  r—^ 
d  tan  7  +  e  sin  7 

Dividing  both  numerator  and  denominator  by  e  tan  7,  we  have; 

sin  7 


Solving  for  tan  ap,  we  have :  tan  a 
rator  ai 
tan  OLP 


d      sin  7 
e     tan  7 

7V  N  sin 

Since    d  =  — £  and  e  =  — ^ ,  and  since  —  =  cos,  we  have: 
2  P  2  P  tan 


tana  = 


sin  7 


COS7 


RULES   AND   FORMULAS 


209 


Formula  (29)  is  derived  by  the  same  process  for  the  other  gear. 
Formulas  (31)  and  (33)  are  derived  from  Fig.  14,  using  the  fol- 
lowing fundamental  equation: 

e       '  d e 

tan  ap      sin  (180°  —  7)      tan  (180°  —  7) 

When  solved  for  tan  ap,  this  gives  Formula  (31). 

Rule  (32),  of  course,  simply  expresses  the  operation  of  finding 
whether  the  pitch  cone  angle  of  the  gear  is  less  than,  equal  to,  or 
greater  than,  90  degrees.  The  derivation  of  Formula  (34)  is 
shown  in  Fig.  15: 

e     N 


Machinery 


Fig.  15.    Diagram  for  Obtaining  Pitch  Cone  Angle  of  Pinion  to  Mesh 
with  Crown  Gear 

D'  . 

Since  in  a  crown  gear  the  dimension  —  in  Fig.  12  is  to  be 

2 

measured  parallel  to  the  axis,  and  will  therefore  be  of  infinite 
length,  the  form  of  the  teeth  will  correspond  to  those  of  a  spur 
gear  having  a  radius  of  infinite  length,  that  is  to  say,  to  a  rack. 
This  accounts  for  Formula  (38). 

Formulas  (39),  (40),  (42)  and  (44)  are  simply  the  correspond- 
ing Formulas  (33),  (9),  (16)  and  (20)  changed  to  avoid  the  use 
of  negative  cosines,  etc.,  which  occur  with  angles  greater  than 
90  degrees.  These  negative  functions  might  possibly  confuse 
readers  whose  knowledge  of  trigonometry  is  elementary.  The 
other  formulas  for  internal  gears  are  readily  comprehended  from 
an  inspection  of  Fig.  n. 


2IO 


BEVEL   GEARING 


Rules  and  Formulas  for  Calculating  Bevel  Gears  with  Shafts  at 
Right  Angles 


^dffW.       aP=PofCpin?onn-an8le     I*"    ™*yJT^^~* 

^ 

m 

spr/v 

)y  NP=i 

.  —  •' 

se  Rules  and  Forn 

angle  of                          x^3  1   l^^fk    T 

gear,              &  ViX^\\\\\<^  ~h 
Number  of  ^    T  /X^^}^ 
teeth  in       tsffic      E 

f^W^ 

t 

u 

pinion,  etc.       *?^  }<  ---D  >i  ^  ^~ 
lulas  Nos.  i  to  21  in  the  order  given. 

No. 

To  Find 

Rule 

Formula 

i 

Pitch  Cone  Angle 
(or  Edge  Angle)  of 
Pinion. 

Divide    the    number    of 
teeth  in  the  pinion  by  the 
number  of  teeth  in  the  gear 
to  get  the  tangent. 

Np 

^           Ng 

2 

Pitch  Cone  Angle 
(or  Edge  Angle)  of 
Gear. 

Divide    the    number    of 
teeth  in  the  gear  by  the 
number  of  teeth  in  the  pin- 
ion to  get  the  tangent. 

—  g 

3 

Proof  of  Calcula- 
tions for  Pitch  Cone 
Angles. 

The    sum    of    the    pitch 
cone  angles  of  the  pinion 
and  gear  equals  90  degrees. 

Ctp  -{-  OLg    =    9O  ° 

4 

Pitch  Diameter. 

Divide    the    number    of 
teeth    by    the    diametral 
pitch;  or  multiply  the  num- 
ber of  teeth  by  the  circular 
pitch  and  divide  by  3.1416. 

D  _  N  _  NP' 

5 

1  These  dimensions  are  the  same  for  both 
gear  and  pinion. 

Addendum. 

Divide  i.o  by  the  diam- 
etral pitch;  or  multiply  the 
circular  pitch  by  0.318. 

~      i.o 

"  p 
=  0.318  P' 

6 

Dedendum. 

Divide  1.157  by  the  diam- 
etral pitch;  or  multiply  the 
circular  pitch  by  0.368. 

=0.368  P' 

7 

Whole  Depth 
of  Tooth  Space. 

Divide  2.157  by  the  diam- 
etral pitch;  or  multiply  the 
circular  pitch  by  0.687. 

TT7          2-I57 

w      p 

=  0.687  P' 

8 

Thickness    of 
Tooth  at  Pitch 
Line. 

Divide  1.571  by  the  diam- 
etral pitch;    or  divide  the 
circular  pitch  by  2. 

T    1.571     P' 

1              P               2 

9 

Pitch      Cone 
Radius. 

Divide   the  pitch   diam- 
eter by  twice  the  sine  of  the 
pitch  cone  angle. 

c-     D 

2  X  sin  a 

10 

Addendum  of 
Small    End    of 
Tooth. 

Subtract    the    width    of 
face  from  the  pitch  cone 
radius,      divide     the     re- 
mainder by  the  pitch  cone 
radius  and  multiply  by  the 
addendum. 

C-F 

° 

RULES  AND   FORMULAS 


211 


Rules  and  Formulas  for  Calculating  Bevel  Gears  with  Shafts  at 
Right  Angles 


No. 

To  Find 

Rule 

Formula 

II 

These  dimensions  are  the  same 
for  both  gear  and  pinion. 

Thickness  of 
Tooth  at  Pitch 
Line   at  Small 
End. 

Subtract  the  width  of 
face  from  the  pitch  cone 
radius,  divide  the  re- 
mainder by  the  pitch  cone 
radius  and  multiply  by  the 
thickness  of  the  tooth  at 
the  pitch  line. 

t      TXC~F 

-ix    c 

12 

Addendum 
Angle. 

Divide  the  addendum  by 
the  pitch  cone  radius  to  get 
the  tangent. 

tan*  =  § 

13 

Dedendum 
Angle. 

Divide  the  dedendum  by 
the  pitch  cone  radius  to  get 
the  tangent. 

ndl      S  +  A 

lan  <p  —       -, 

14 

Face  Angle. 

Subtract  the  sum  of  the 
pitch  cone  and  addendum 
angles  from  90  degrees. 

d  =  go°-(a  +  e) 

15 

Cutting  Angle.* 

Subtract  the  dedendum 
angle  from  the  pitch  cone 
angle. 

r=«-0 

16 

Angular     Adden- 
dum. 

Multiply  the  addendum 
by  the  cosine  of  the  pitch 
cone  angle. 

K=SXcosa 

17 

Outside  Diameter. 

Add  twice  the  angular 
addendum  to  the  pitch 
diameter. 

O  =  D  +  2K 

18 

Apex  Distance. 

Multiply  one-half  the 
outside  diameter  by  the 
tangent  of  the  face  angle. 

J  =  -  X  tan  5 

19 

Apex        Distance 
at    Small    End    of 
Tooth. 

Subtract  the  width  of  face 
from  the  pitch  cone  radius; 
divide  the  remainder  by 
the  pitch  cone  radius  and 
multiply  by  the  apex  dis- 
tance. 

i     JXC~F 

J-JX      c 

20 

Number  of  Teeth 
.for  which  to  Select 
Cutter. 

Divide  the  number  of 
teeth  by  the  cosine  of  the 
pitch  cone  angle. 

N'        N 

cos  a 

21 

Proof  of  Calcula- 
tions by  Rules  Nos. 
9,  12,  14,  16  and  17. 

The  outside  diameter 
equals  twice  the  pitch  cone 
radius  multiplied  by  the 
cosine  of  the  face  angle  and 
divided  by  the  cosine  of  the 
addendum  angle. 

^       2CXcos5 

COS0 

•  See  Chapter  XIII,  paragraph  "  Cutting  Bevel  Gears  in  the  Milling  Machine." 


212 


BEVEL   GEARING 


Rules  and  Formulas  for  Calculating  Miter  Bevel  Gearing 


Use  Rules  and  Formulas  Nos.  22,  4-8,  23,  10-13,  24-26,  17-19,  27  and  21 
in  the  order  given.  All  dimensions  thus  obtained  are  the  same  for  both 
gears  of  a  pair. 


No. 


22 


To  Find 


Rule 


Formula 


Pitch  Cone  Angle. 


Pitch  cone   angle   equals 
45  degrees. 


«=4S 


23 


Pitch  Cone  Radius. 


Multiply  the  pitch  diam- 
eter oy  0.707. 


£=0.707  D 


24 


Face  Angle. 


•  Subtract    the    addendum 
angle  from  45°. 


25 


Cutting  Angle.* 


Subtract    the    dedendum 
angle  from  45°. 


26 


Angular  Addendum. 


Multiply    the    addendum 
by  0.707. 


#=0.7075 


27 


Number  of  Teeth 
for  which  to  Select 
Cutter. 


Multiply   the   number  of 
teeth  by  1.41. 


See  Chapter  XIII,  paragraph  "  Cutting  Bevel  Gears  in  the  Milling  Machine. 


Examples  of  Bevel  Gear  Calculations.  —  A  number  of  examples 
of  calculations  are  given  in  the  following  for  practice,  covering 
all  the  various  types  shown  in  Figs.  3  to  8  inclusive.  The  con- 
ditions of  the  various  examples  differ  from  each  other  only  in  the 
center  angle.  While  such  great  accuracy  is  not  required  in  the 
work  itself,  it  will  be  found  convenient  in  the  calculations  to 
use  tables  of  sines  and  tangents  which  give  readings  for  minutes 
to  five  figures.  This  permits  accurate  checking  of  the  various 
dimensions  by  Rules  and  Formulas  (3),  (21),  etc. 

Shafts  at  Right  Angles.  —  Let  it  be  required  to  make  the 
necessary  calculations  for  a  pair  of  bevel  gears  in  which  the  shafts 
are  at  right  angles;  diametral  pitch  =  3,  number  of  teeth  in 
gear  =  60,  number  of  teeth  in  pinion  =  15,  and  width  of  face 
=  4  inches. 


RULES  AND   FORMULAS 


213 


Rules  and  Formulas  for  Calculating  Bevel  Gears  with  Shafts  at  an 

Acute  Angle 


ctp=  Pitch  cone  angle  of           VERTE 
^i*\~  —  ffc^L                  pinion;                                  / 
/    !  \  7^%      «„=  Pitch  cone  angle  of            £* 
^—  —  1  —  ^L^^f^^               ffear:                -x           .././  ^ 

°V^  H 
K?\TT--^  f~T» 

sk          C*     >i  K-K 

ctw   "^>\.        \J  i         1 
~      1     \ALi      >l         .-, 

-»    v     'f^>Oxfyij.  1 

t&ffpSyy'f^ft 

K—  -  "'" 
Use  Rules  and  Formulas 

Jumber     of  Jr   N  .  ///faxm 
teeth  in      ^  ^$^y 
pinion,         %$/*?}             b 

TBll^ 

etc.                   **•*—           -°        —*">*" 
>  Nos.  28-30,  and  4-21  in  the  order  given. 

No. 

To  Find 

Rule 

Formula 

28 

Pitch  Cone  Angle 
(or  Edge  Angle)  of 
Pinion. 

Divide  the  sine  of  the 
center  angle  by  the  sum 
of  the  cosine  of  the  cen- 
ter angle  and  the  quo- 
tient of  number  of  teeth 
in  the  gear  divided  by 
the  number  of  teeth  in 
the  pinion;  this  gives  the 
tangent. 

sin  y 

^  +  cosr 
Np 

29 

Pitch  Cone  Angle 
(or  Edge  Angle)  of 
Gear. 

Divide  the  sine  of  the 
center  angle  by  the  sum 
of  the  cosine  of  the  cen- 
ter angle  and  the  quo- 
tient of  the  number  of 
teeth  in  the  pinion  di- 
vided by  the  number  of 
teeth  in  the  gear;    this 
gives  the  tangent. 

sin  7 

tana<,  -  —  - 
]J+cos7 

30 

Proof  of  Calcula- 
tions for  Pitch  Cone 
Angles. 

The  sum  of  the  pitch 
cone  angles  of  the  pin- 
ion and  gear  equals  the 
center  angle. 

Otp  +  OCg  —  J 

tan  ap  =  15  -T-  60  =  0.25000  =  tan  14°  2'.    .     .  (i) 

tan  a.g  =  60  -T-  15  =  4.00000  =  tan  75°  58'  .     .  (2) 

7  =  14°  2'  +  75°  58'  =  90° (3) 

DP  =  is  +  3  =  s-ooo"   .  .  ,r  .  .:....  (4) 

5  =  1^-3  =  0.3333".  . •.."'.   <  .    •  .  (5) 

=  0.3856".     .    .   v  (6) 


W  =  ^^  =  0.7190' 
o 


iSH-  0.5236". 

o 


(7) 
(8) 


214 


BEVEL   GEARING 


Rules  and  Formulas  for  Calculating  Bevel  Gears  with  Shafts  at  an 
Obtuse  Angle 


aP  =  V 

JjftV-' 

itch  cone  angle                           /\ 
of  pinion  ;                          VERTEX  -^-.     \^  -^ 
itch  cone  angle                     /'  I  X       °-\*\  *-K  T 

T°f  lear;  f                        XT!  t\^2tt     7 
lumber  of          v          ^7    H  v  -.^A^y.  . 

H«U  i  Kfe^Sfo--/-^  -^rp=^ 

flf^-                   teeth  in       /\|^« 
^-f^    -/ji                    pinion,          \    ^W__ 

>>x        etc-      v^r  — 

H-~'                                    H  

Use  Rules  and  Formulas  Nos.  31  and  32  as  dir 

ITfm^A 

IJJjSj 

:E-ffi 

ected  below. 

No. 

To  Find 

Rule 

Formula 

31 

Pitch  Cone  Angle 
(or  Edge  Angle)  of 
Pinion. 

Divide  the  sine  of  180 
degrees  minus  the  center 
angle  by  the  difference 
between  the  quotient  of 
the  number  of  teeth  in 
the  gear  divided  by  the 
number  of  teeth  in  the 
pinion  and  the  cosine  of 
1  80  degrees  minus  the 
center  angle;  this  gives 
the  tangent. 

tan«p  = 

sin  (180°—  7) 

^-cos(i8o°-7) 

l\p 

32 

Whether  Gear  is  a 
Regular  Bevel  Gear, 
a  Crown    Gear,    or 
an    Internal    Bevel 
Gear. 

Add  90  degrees  to  the 
pitch  cone  angle  of  the 
pinion.     If  the   sum  is 
greater  than  the  center 
angle    use    Rules    and 
Formulas    Nos.    33,    30 
and  4-21    in   the   order 
given.       If     the      sum 
equals  the  center  angle 
see  rules  and  formulas 
for  crown  gear.     If  the 
sum  is  less  than  the  cen- 
ter angle  see  rules  and 
formulas     for     internal 
bevel  gear. 

33 

Pitch  Cone  Angle 
(or  Edge  Angle)  of 
Gear. 

Divide  the  sine  of  180 
degrees  minus  the  center 
angle  by  the  difference 
between  the  quotient  of 
the    number    of    teeth 
in  the  pinion  divided  by 
the  number  of  teeth  in 
the  gear  and  the  cosine 
of  1  80  degrees  minus  the 
center  angle;   this  gives 
the  tangent. 

tanaff= 
sin  (180°—  7) 

^?-cos(i8o°-7) 

JMg 

RULES  AND   FORMULAS 


215 


Rules  and  Formulas  for  Calculating  Crown  Gears 


$<K=ZERO 

t^     o D  ±\          I       o 

VERTEX     lT"V~2'|^2  ~~    . 


a  =90 


\ 

'  %  «p=Pitch  cone  angle  of    !        ^^  ^  I  S+A 

^  pinion; 

'      ^Vp= Number  of  teeth  in 

pinion; 

I*""  Ng  =  Number  of  teeth  in 

gear,  etc. 

Use  Rules  Nos.  31  and  4-21  in  the  order  given,  for  the  pinion;  use 
Rules  Nos.  30,  4-8,  36, 10-13,  37, 15  and  38  in  the  order  given  for  the  crown 
gear;  if  dimensions  for  crown  gear  are  known,  to  find  center  angle  and 
dimensions  of  pinion,  use  Rules  and  Formulas  Nos.  34,  35  and  4-21  in 
the  order  given. 


No. 


To  Find 


Rule 


Formula 


34 


Pitch  Cone  Angle 
(or  Edge  Angle)  of 
Pinion. 


Divide  the  number  of 
teeth  in  the  pinion  by  the 
number  of  teeth  in  ,the 
gear,  to  get  the  sine. 


N 


35 
36 


Center  Angle. 


Add  90  degrees  to  the  pitch 
cone  angle  of  the  pinion. 


Pitch  Cone 
Radius. 


Divide  the  pitch  diam- 
eter by  2. 


X/A      LJJ      4  • 

The  face  cone  angle  of 
the  gear  equals  the  adden- 
dum angle. 


37 


Face  Angle  of 
Gear. 


Number  of  Teeth 
for  which  to  Select 
Cutter. 


The  teeth  are  equivalent 
in  form  to  rack  teeth. 


Ng'  —  infinity 


c  = 


2  X  0.24249 

6.1 

=  0-3333 


10.3097". 

=  0.2040' 


/  =  0.5236  X  -^_  =0.3204"    . 
10.31 

tan  e  =  °'3333   =  0.03233  =  tan  i°  51'    . 
10.3097 

0.3856  o    , 

tan  (j>  =  — °  J     =  0.03740  =  tan  2  9  .     . 
10.3097 


(9) 
(10) 


•  (12) 

•  (i3) 
(14) 


2l6  BEVEL  GEARING 

Rules  and  Formulas  for  Calculating  Internal  Bevel  Gears 


^m 

A, 

o/      § 

ff-4^  °  ~^*V~K 

t&dL     ,  &&M£ 

f  SfiSsj     *}    SfeX&r 

ce  angle  of  gear;                £ 
imber  of  teeth  in       *  
)inion; 
mber  of  teeth  in 
;ear,  etc. 

^os.  31  and  4-21  inclusive  f 
>,  30,  40,41,  15,  42,43»  18,  i 

^^^W^j    v 

vi-_i^RT~rr 

**  1  ™          &  £    6g—  ±<a 

jf  y^4=NL 

L  •  •"     ^^                   ] 

±^r       ^=NU 
* 

Use  Rules  and  Formulas  I 
Rules  and  Formulas  Nos.  3c 
order  given  for  the  gear. 

^j*  -H 

or  the  pinion;  use 
9,  44  and  21  in  the 

No. 

To  Find 

Rule 

Formula 

39 

Pitch  Cone  Angle 
(or  Edge  Angle)  of 
Gear. 

Divide  the  sine  of  180 
degrees  minus  the  center 
angle  by  the  difference 
between   the   cosine   of 
1  80  degrees  minus  the 
center    angle    and    the 
quotient  of  the  number 
of  teeth  in  the  pinion 
divided  by  the  number 
of  teeth  in  the  gear;  sub- 
tract the   angle  whose 
tangent   is   thus  found 
from  1  80  degrees. 

tanaa  = 
sin  (180  —  7) 

CO.  (!&,-„)-§ 

ag  =  180  —  OLa 

40 

Pitch  Cone 
Radius. 

Divide  the  pitch  diam- 
eter by  twice  the  sine  of 
1  80  degrees  minus  the 
pitch  cone  angle. 

c-        D° 

2  sin  (i  80  —  ag] 

4i 

Face  Angle  of 
Gear. 

Subtract    90    degrees 
from    the    sum    of    the 
pitch  cone  angle  and  the 
addendum  angle. 

8g  =  <xg  +  d-90° 

42 

Angular  Adden- 
dum of  Gear. 

Multiply   the   adden- 
dum by  the  cosine  of 
1  80  degrees  minus  the 
pitch  cone  angle. 

Kg  = 

S  X  cos  (180  —  ag} 

43 

Outside  (or  Edge) 
Diameter  of  Gear. 

Subtract     twice     the 
angular  addendum  from 
the  pitch  diameter. 

Qg   =    Dg   —    2  Kg 

44 

Number  of  Teeth 
for  which  to  Select 
Cutter. 

Divide  the  number  of 
teeth  by  the  cosine  of 
1  80  degrees  minus  the 
pitch  cone  angle. 

N'              N° 

COS  (l8o  —  <Xg, 

RULES  AND   FORMULAS  217 

K  =  0.3333  X  0.97015  =  0.3234"   .     .     .     (16) 
0  =  5.000  +  2X0.3234  =5.6468".     .    .     (17) 


j  =  9.9225  X  —  ^i-  =  6.0726"  ....     (19) 

"O 

=  15.4    .  .  I  /';,,  .  (20) 


0.97015 
5.6468"  ^ 


0.99948 

This  gives  all  the  data  required  for  the  pinion.  Rules  (5)  to 
(13)  inclusive  apply  equally  to  the  gear  and  the  pinion,  so  we 
have  only  calculations  by  Rules  and  Formulas  (4)  and  (14)  to 
(21)  to  make,  though  it  is  well  to  calculate  Formula  (9)  a  second 
time  as  a  check  for  the  same  calculation  for  the  pinion. 

D  =  —  =  20.000"   ....         ...      (4) 

o 

=  10.3077".  (9) 


2  X  0.97015 

f  =  75°  58/  -  2°  9'  =  73°  49'.    ....  .  .  (15) 

K  =  °-3333  X  0.24249  =  0.0808"    .  .  .  (16) 

0  =  20  +  2  X  0.0808  =  20.1616"  .  .  •'.  (17) 

T      20.1616  v                          ,  „  f  0. 

J  =  — X  0.2159  =  2.1764      .  .  .  (18) 


j  =  2.1764 x ^-  =  1.3320"  .  v,..  .  (19) 

ff' --*>-=  247  .  (20) 

0.24249 

2o.i6i6"^2°-6l54X°-97748=20l6  „      (2I) 
0.99948 

This  gives  the  calculations  necessary  for  this  pair  of  gears, 
which  are  shown  dimensioned  in  Fig.  i,  Chapter  XI.  There  are 
two  or  three  other  dimensions,  such  as  the  over-all  length  of  the 
pinion,  etc.,  which  depend  on  arbitrary  dimensions  given  the 


2l8  BEVEL   GEARING 

gear  blank.     Directions  for  calculating  these  are  given  in  Chapter 
XI  in  connection  with  Fig.  i  of  that  chapter. 

Acute  Angle  Bevel  Gearing.  —  Let  it  next  be  required  to  cal- 
culate the  dimensions  of  a  pair  of  bevel  gears  whose  center  angle 
is  75  degrees,  the  number  of  teeth  in  the  pinion  15,  the  number  of 
teeth  in  the  gear  60,  the  diametral  pitch  3,  and  the  width  of  face 
4  inches.  This  is  the  same  as  the  first  example,  except  for  the 
center  angle.  Following  the  directions  given  in  the  chart  we  have  : 

°47'.    .     (28) 


-  tan  62*13'     -     (29) 
T  =  12°  47'  +  62°  13'  =  75°    .     .     .    .    <    .     (30) 

Formulas  (4)  to  (8)  as  in  first  example;  also,  C  =  11.2989", 
s  =  0.2154",  /  =  0.3382",   e  =  i°4i',   0  =  i°  57',  5  =  75°  32', 

r  =  10°  50',  K  =  0.3251",  o  =  5.6502",  /  =  10.9501",  j  = 

7.0748",  and  N'  =  15.3,  also, 


For  the  gear,  the  additional  calculations  give:  C  —  11.303", 
8  =  26°  6',  r  =  60°  16',  K  =  0.1553",  0  =  20.3106",  /  =  4.9748", 
j  =  3.2142",  Nf  =  129. 

,,,  ^  22.606  X  0.89803  ,tt  f    N 

20.3106    ^  -  2  —  2*  =  20.3096"    .     .     (21) 

0-99957 

The  above  calculations  are  not  all  given  in  full,  as  most  of 
them  are  merely  re-duplications  of  formulas  previously  used. 

Obtuse  Angle  Bevel  Gearing.  —  Let  it  be  required  to  calculate 
the  dimensions  of  the  same  set  of  gears  but  with  the  center  angle 
of  100  degrees.  This  being  an  example  of  obtuse  angle  gearing, 
we  apply  Formula  (31)  as  follows: 


tan  ap  =  jLo!          =  °'25738  =  tan  14°  26'     .     (31) 

s  discover  that  it  is  an  example  of  regular  obtuse  angle 
since 

14°  26'  +  90°  =  104°  26'  >  100°  .....     (32) 


RULES  AND   FORMULAS  219 

The  remaining  calculations  for  the  angles  are  as  follows: 

tan  a  =      °fQ^6s  =  12.8986  =  tan  85°  34' .     -     (33) 

y  =  14°  26'  +  85°  34'  =  100° (30) 

and  the  calculations  for  the  other  dimensions  as  per  the  table. 

Crown  Gear.  —  Suppose  it  is  required  to  make  a  crown  gear 
and  a  pinion  for  the  same  number  of  teeth,  pitch  and  face  as  in 
the  previous  example.  What  are  the  additional  calculations 
necessary?  Following  the  proper  formulas  in  the  order  given 
by  the  chart,  we  have: 

sin  ap  =  -—  =  0.25000  =  sin  14°  29'.     .     .     .     (34) 
oo 

7  =  90°  +  14°  29'  =  104°  29'     ....     (35) 

The  other  calculations  are  similar  to  those  already  given. 

Internal  Bevel  Gear.  —  Let  it  be  required  to  design  a  pair  of 
bevel  gears  of  the  same  number  of  teeth,  pitch  and  face,  in  which 
the  center  angle  is  115  degrees.  This  being  an  example  of  obtuse 
angle  gearing,  we  use  Formula  (31): 

0.00631  o     /  /    \ 

tan  ap  =  ^-5—        —  =  0.25334  =  tan  14  13  .     .     (31) 
T5  ~~  0.42202 

Thus,  according  to  Rule  (32),  we  find  that 

14°  13'  +  90°  =  104°  13' <  "5° (32) 

showing  that  the  gear  is  an  internal  bevel  gear.    Applying  the 
rules  and  formulas  for  internal  bevel  gearing,  we  have: 

79°  I3/ 

.    .    .    .    .    (39) 
7  =  100°  47' +  14°  13' =  115°  ...    .    .    .    .     (30) 

^2O  „  ,        . 

^  =  ~T: — o —  =  10.1797" (40) 

2  X  0.98234 

8  =  100°  47'  +1°  53'  -90°  =12°  40' (41) 

f  =  98°  37X,  and  #  =  0.0624" 

0  =  20  —  2  X  0.0624  =   19.8752" (43) 

=  320  (internal).    .    .    .    .    ;    .    .     (44) 


220 


BEVEL   GEARING 


The  calculations  for  the  pinion  and  the  other  calculations  for 
the  gear  are  similar  to  those  already  given. 

How  to  Avoid  Internal  Bevel  Gears.  —  When  Rule  (32),  in 
any  given  case,  shows  that  the  large  gear  will  be  an  internal  bevel 
gear,  such  as  shown  in  Fig.  16,  this  construction  may  be  avoided 
without  changing  the  position  of  the  shafts,  the  numbers  of  the 
teeth  in  the  gear,  the  pitch  of  the  teeth,  or  the  width  of  face. 
This  is  done  simply  by  subtracting  the  given  center  angle  from 
1 80  degrees,  and  using  the  remainder  as  a  new  center  angle  in 
calculating  a  set  of  acute  angle  gears  by  Rules  and  Formulas  (28), 
(29),  (30),  etc.  A  pair  of  bevel  gears  calculated  on  this  basis 


Machinery 


Fig.  16.    Internal  Bevel  Gearing, 
a  Type  to  be  Avoided 


Fig.  17.    Acute-angle  Bevel  Gears 
substituting  Internal  Gearing 


corresponding  to  those  in  Fig.  16  are  shown  in  Fig.  17.  It  will 
be  seen  that  the  contact  takes  place  on  the  other  side  of  the  axis 
OP  of  the  pinion. 

It  is  necessary  to  avoid  internal  bevel  gears  as  they  can  only 
be  used  with  cast  teeth,  it  being  practically  impossible  to  cut 
them.  It  may  be  that  some  forms  of  templet  planing  machines 
will  do  this  work,  if  the  pitch  cone  angle  is  not  too  great,  but  no 
form  of  generating  machine  will  do  it. 

Systems  of  Tooth  Outlines  used  for  Bevel  Gearing.  —  Five 
systems  of  tooth  outlines  are  commonly  used  for  bevel  gearing. 
They  are  the  cycloid,  the  standard  i4^-degree  involute,  the  20- 
degree  involute  and  the  15-  and  2o-degree  octoid. 


RULES  AND   FORMULAS  221 

The  cycloidal  form  of  tooth  is  obsolete  for  cut  bevel  gears,  and 
is  rarely  met  with  nowadays  even  for  cast  gears.  It  requires  very 
careful  workmanship,  and  is  difficult  or  impossible  to  generate. 
It  is  also  a  bad  shape  to  form  with  a  relieved  cutter,  as  the  cutting 
edge  tends  to  drag  at  the  pitch  line,  where  for  a  short  distance 
the  sides  of  the  teeth  are  nearly  or  quite  parallel.  For  spur 
gearing  it  has  a  few  points  of  advantage  over  the  involute  form 
of  tooth',  but  in  the  case  of  bevel  gearing  these  are  nullified  by 
the  impossibility  of  generating  the  teeth  in  practicable  machines. 
The  cycloidal  form  of  tooth  need  not  be  seriously  considered  for 
bevel  gears. 

Involute  Teeth.  —  Most  bevel  gears  are  made  on  the  involute 
system,  of  either  the  standard  i4f-degree  pressure  angle,  or  the 
2o-degree  pressure  angle.  In  spur  gear  teeth  the  pressure  angle 
may  be  defined  as  the  angle  which  the  flat  surface  of  the  rack 
tooth  makes  with  the  perpendicular  to  the  pitch  line.  The  20- 
degree  tooth  is  consequently  broader  at  the  base  and  stronger 
in  form  than  the  i4§-degree  tooth.  This  same  difference  applies 
to  bevel  gears.  Most  bevel  gears  that  are  milled  with  formed 
cutters  are  made  to  the  i^-degree  standard,  as  cutters  for  this 
shape  are  regularly  carried  in  stock.  The  planed  gears,  made  by 
the  templet  or  generating  principles,  are  nowadays  often  made 
to  the  2o-degree  pressure  angle,  both  for  the  sake  of  obtaining 
stronger  teeth,  and  for  avoiding  undercutting  of  the  flanks  of  the 
pinions  as  well.  This  undercutting  is  due  to  the  phenomenon  of 
"  interference/'  as  it  is  called,  which  is  minimized  by  increasing 
the  pressure  angle. 

Octoid  Teeth.  —  If  a  manufacturer  is  asked  to  plane  a  pair  of 
involute  bevel  gears  on  the  Bilgram,  Gleason  or  other  similar 
generating  machine,  he  will  not  produce  involute  teeth,  but 
something  "just  as  good."  This  "just  as  good"  form  was  in- 
vented by  Mr.  Bilgram,  and  was  named  "Octoid"  by  Mr.  Geo. 
Grant.  In  generating  machines  the  teeth  of  the  gears  are  shaped 
by  a  tool  which  represents  the  side  of  the  tooth  of  an  imaginary 
crown  gear.  The  cutting  edge  of  the  tool  is  a  straight  line,  since 
the  imaginary  crown  gear  has  teeth  whose  sides  are  plane  sur- 
faces. It  can  be  shown  that  the  teeth  of  a  true  involute  crown 


222 


BEVEL   GEARING 


gear  have  sides  which  are  very  slightly  curved.  The  minute 
difference  between  the  tooth  shapes  produced  by  a  plane  crown 
tooth  and  a  slightly  curved  crown  tooth  is  the  minute  difference 
between  the  octoid  and  involute  forms.  Both  give  theoretically 
correct  action.  The  customer  in  ordering  gears  never  uses  the 
word  " octoid,"  as  it  is  not  a  commercial  term;  he  calls  for 
"involute"  gears. 

Formed  Cutters  for  Involute  Bevel  Gear  Teeth.  —  For  14^- 
degree  involute  teeth,  the  shapes  of  the  standard  cutter  series 
furnished  by  the  makers  of  formed  gear  cutters  for  bevel  gears 
are  commonly  used.  There  are  8  cutters  in  the  series,  to  cover 
the  full  range  from  the  i2-tooth  pinion  to  a  crown  gear.  The 
various  cutters  are  numbered  from  i  to  8,  as  given  in  the  table 
below. 

Formed  Cutters  for  Involute  Bevel  Gear  Teeth 


Number  of 
Cutter 

Number  of  Teeth 
for  which  Cutter 
is  Used 

Number  of 
Cutter 

Number  of  Teeth 
for  which  Cutter 
is  Used 

I 
2 

3 

4 

From  135  to  a  Rack 
From    55  to  134 
From    35  to    54 
From    26  to    34 

6 

7 
8 

From  21  to  25 
From  17  to  20 
From  14  to  16 
From  12  to  13 

It  should  be  remembered  that  the  number  of  teeth  in  this  table 
refers  to  the  number  of  teeth  in  the  equivalent  spur  gear,  as 
given  by  Rule  (20),  which  should  always  be  used  in  selecting  the 
cutter  used  for  milling  the  teeth  of  bevel  gears.  The  standard 
bevel  gear  cutter  is  made  thinner  than  the  standard  spur  gear 
cutter,  as  it  must  pass  through  the  narrow  tooth  space  at  the 
inner  end  of  the  face.  As  usually  kept  in  stock,  these  cutters 
are  thin  enough  for  bevel  gears  in  which  the  width  of  face  is  not 
more  than  one- third  the  pitch  cone  radius.  Where  the  width  of 
face  is  greater,  special  cutters  have  to  be  made,  and  the  manu- 
facturer should  be  informed  as  to  the  thickness  of  the  tooth  space 
at  the  small  end;  this  will  enable  him  to  make  the  cutter  of  the 
proper  width. 


RULES  AND   FORMULAS  223 

Special  Forms  of  Bevel  Gear  Teeth.  —  In  generating  machines 
(such  as  the  Bilgram  and  the  Gleason)  it  is  often  advisable  to 
depart  from  the  standard  dimensions  of  gear  teeth  as  given  by 
Rules  and  Formulas  (i)  to  (44).  For  instance,  where  the  pinion 
is  made  of  bronze  and  the  gear  of  steel,  the  teeth  of  the  former 
can  be  made  wider  and  those  of  the  latter  correspondingly 
thinner,  so  as  to  somewhere  nearly  equalize  the  strength  of  the 
two.  Again,  where  the  pinion  has  few  teeth  and  the  gear  many, 
it  may  be  advisable  to  make  the  addendum  on  the  pinion  larger 
and  the  dedendum  correspondingly  smaller,  reversing  this  on  the 
gear,  making  the  addendum  smaller  and  the  dedendum  larger. 
This  is  done  to  avoid  interference  and  consequent  undercut  on 
the  flanks  of  pinions  having  a  small  number  of  teeth.  Such 
changes  are  easily  effected  on  generating  machines  and  instruc- 
tions for  doing  this  for  any  case  will  be  furnished  by  the  makers. 


CHAPTER  XI 
STRENGTH   AND   DESIGN   OF    BEVEL   GEARS 

The  Materials  Used  for  Making  Bevel  Gears. —The  same 
materials  are  used  in  general  for  making  bevel  gears  as  for  spur 
gears  and  each  has  practically  the  same  advantages  and  dis- 
advantages for  both  cases.  In  general,  the  strength  of  different 
materials  is  roughly  proportional  to  the  durability. 

Cast  iron  is  used  for  the  largest  work,  and  for  smaller  work 
which  is  not  to  be  subjected  to  heavy  duty.  In  cases  where  great 
working  stress  or  a  sudden  shock  is  liable  to  come  on  the  teeth, 
steel  is  ordinarily  used.  Such  gears  are  made  from  bar  stock  for 
the  smallest  work,  from  drop  forgings  for  intermediate  sizes  made 
on  a  manufacturing  basis,  and  from  steel  castings  for  heavy  work. 
The  softer  grades  of  steel  are  not  fitted  for  high-speed  service,  as 
this  material  abrades  more  rapidly  than  cast  iron.  This  objec- 
tion does  not  apply  to  hardened  steels,  such  as  used  in  automobile 
transmission  gears. 

As  in  the  case  of  spur  gearing  it  is  quite  common  to  make  the 
gear  and  pinion  of  different  materials.  TJiis  is  advantageous 
from  the  standpoint  of  both  efficiency  and  durability,  since  two 
dissimilar  metals  work  on  each  other  with  less  friction  than  simi- 
lar metals,  as  is  well  known.  Cast  iron  and  steel,  and  steel  and 
bronze  are  common  combinations.  In  general,  the  pinion  should 
be  made  of  the  stronger  material,  since  it  is  of  weak  form;  and 
it  should  be  made  of  the  more  durable  material,  as  it  revolves 
more  rapidly  and  each  tooth  comes  into  working  contact  more 
times  per  minute  than  do  those  of  the  larger  mating  gear.  In 
a  steel  and  cast  iron  combination,  then,  the  pinion  should  be 
of  steel,  while  the  gear  is  of  cast  iron.  In  a  steel  and  bronze 
combination,  the  pinion  should  be  of  steel  and  the  gear  of 
bronze,  though  this  is  more  costly  than  when  the  materials  are 
reversed. 

224 


STRENGTH  AND   DESIGN  225 

A  wide  range  of  physical  qualities  is  now  available  in  steel,  both 
for  parts  small  enough  to  be  made  from  bar  stock,  and  for  those 
made  from  drop  forgings.  Recent  improvements  have  also  given 
almost  as  much  flexibility  in  the  choice  of  steel  castings.  Gears 
made  from  high-grade  steels  may  be  subjected  to  heat-treatments 
which  increase  their  durability  and  strength  amazingly. 

Rawhide  and  fiber  are  quite  largely  used  for  pinion  blanks  in 
cases  where  it  is  desired  to  run  gearing  at  a  very  high  speed  and 
with  as  little  noise  as  possible.  There  is  a  little  more  difficulty 
in  building  up  a  rawhide  blank  properly  for  a  bevel  gear  than 
for  a  spur  gear.  Fiber,  which  is  used  in  somewhat  the  same  way, 
has  the  merit  of  convenience  and  comparative  inexpensiveness, 
as  it  may  be  purchased  in  a  variety  of  sizes  of  bars,  rods,  tubes, 
etc.,  ready  to  be  worked  up  into  pinion  blanks  at  short  notice. 
It  is  not  so  strong  as  rawhide,  and  is  difficult  to  machine  owing 
to  its  gritty  composition.  For  light  duty  at  high  speed  it  does 
very  well.  For  large,  high-speed  gearing  it  was  formerly  a  com- 
mon practice  to  use  inserted  wooden  teeth  on  the  gear,  meshing 
with  a  solid  cast-iron  pinion.  This  construction  is  seldom  used 
for  cut  gearing. 

Strength  of  Bevel  Gear  Teeth.  —  The  Lewis  formula  is  the 
one  generally  used  for  calculating  the  strength  of  gears.  Mr. 
Myers,  in  an  article  on  the  " Strength  of  Gears"  in  the  December, 
1906,  number  of  MACHINERY,  gives  Mr.  Earth's  adaptation  of 
this  formula  for  calculating  the  strength  of  bevel  gears.  The 
rules  and  formulas  in  the  following  are  condensed  from  the 
method  given  in  the  article  referred  to. 

The  factors  to  be  taken  into  account  are  the  pitch  diameter  of 
the  gear,  the  number  of  revolutions  per  minute,  the  diametral 
pitch  (or  circular  pitch  as  the  case  may  be),  the  width  of  face,  the 
pitch  cone  radius,  the  number  of  teeth  in  the  gear  and  the  maxi- 
mum allowable  static  fiber  stress  for  the  material  used.  From' 
this  may  be  found  the  maximum  allowable  load  at  the  pitch  line, 
and  the  maximum  horsepower  the  gear  should  be  allowed  to 
transmit. 

The  reader  familiar  with  the  Lewis  formula  will  note  that  Rule 
and  Formula  (3)  is  the  same  as  for  spur  gears  with  the  exception 


226 


BEVEL   GEARING 


of  the  additional  factor 


C  -F 


This  factor  is  an  approximate 


one  which  expresses  the  ratio  of  the  strength  of  a  bevel  gear  to 
that  of  a  spur  gear  of  the  same  pitch  and  number  of  teeth,  the 
decrease  being  due  to  the  fact  that  the  pitch  grows  finer  toward 
the  vertex.  This  factor  is  approximate  only  and  should  not  be 
used  for  cases  in  which  F  is  more  than  J  C;  but  since  no  bevel 
gears  should  be  made  in  which  F  is  more  than  |  C,  the  rule  is  of 

Factors  for  Calculating  Strength  of  Bevel  Gears     Pi4.™et»v( 


Ltfrt 

Table  of  Outline  Factors  (Y)  for  14^°  and  20°  Involute 

Outline  Factor  =  Y 

Outline  Factor  =  Y 

^XvVV\ 

N' 

N' 

\    \ 

Involute 
(Std.) 

20° 

Involute 

Involute 
(Std.) 

20° 

Involute 

Y////fy/ 

\       V 

o_ 

zys 

12 

O.2IO 

0.245 

27 

0.314 

0-349 

f/////// 

/ 

13 

O.  22O 

O.  2OI 

30 

0.320 

0.358 

"\VA 

/ 

14 

O.226 

0.276 

34 

0.327 

0.371 

v\ 

15 

0.236 

0.289 

38 

0.336 

0-383 

\/s//h 

16 

O.242 

0.295 

43 

0.346 

0.396 

1  ' 

:sJi 

17 

0.251 

0.302 

50 

0.352 

0.408 

AT/ 

No.  of  teeth 

18 

O.26l 
0.273 

0.308 
0.314 

60 

75 

0.358 
0.364 

0.421 
0-434 

cos  a 

20 

0.283 

0.320 

IOO 

0.371 

0.446 

21 

0.289 

0.327 

150 

0-377 

0-459 

23 

0.295 

0-333 

300 

0.383 

0.471 

25 

0.305 

0-339 

Rack 

0.390 

0.484 

universal  application  for  good  practice.  As  the  width  of  face  is 
made  greater  in  proportion  to  the  pitch  cone  radius,  the  increase 
of  strength  obtained  thereby  grows  proportionately  smaller  and 
smaller,  as  may  be  easily  proved  by  analysis  and  calculation. 
Actually  the  advantage  of  increasing  the  width  of  face  is  even 
less  than  is  indicated  by  calculation,  since  the  unavoidable  de- 
flection and  disalignment  of  the  shaft  is  sure  at  one  time  or 
another  to  throw  practically  the  whole  load  on  the  weak  inner 
ends  of  the  teeth,  which  thus  have  to  carry  the  load  without  help 
from  the  large  pitch  at  the  outer  ends. 

Rules  and  Formulas  for  the  Strength  of  Bevel  Gears.  —  The 
method  for  calculating  the  strength  of  bevel  gearing  is  practi- 


STRENGTH  AND   DESIGN 


227 


cally  the  same  as  that  used  for  spur  gears.  The  accompanying 
tables  of  " Rules  and  Formulas  for  the  Strength  of  Bevel  Gears," 
and  "  Factors  for  Calculating  the  Strength  of  Bevel  Gears,"  in 
combination  with  the  table  "Working  Stresses  used  in  the  Lewis 
Formula  for  the  Strength  of  Gear  Teeth,"  in  Chapter  III, 
Rules  and  Formulas  for  the  Strength  of  Bevel  Gears 


Z)  =  pitch  diameter  of  gear  in  inches;       Y=  outline  factor  (see  table,  op- 
R  =  revolutions  per  minute;                              posite  page); 
V=  velocity  in  ft.  per  min.  at  pitch       P  =  diametral   pitch    (if  circular 
diam.  ;                                                          pitch  is  given,  divide  3  .  1416 
S8  =  allowable  static  unit  stress  for                 by  circular  pitch  to  obtain 
material;                                                     diametral  pitch)  ; 
5=  allowable  unit  stress  for  ma-        C=  pitch  cone  radius; 
terial  at  given  velocity;               W=  maximum  safe  tangential  load 
F=  width  of  face;                                                in  pounds  at  pitch  diam.; 
N'  =  No.  of  teeth  in  equivalent  spur  H.  P.  =  maximum  safe  horsepower, 
gear  (see  diagram  in  table  on 
opposite  page)  ; 

Use  Rules  and  Formulas  Nos.  i  to  4  in  the  order  given. 

No. 

To  Find 

Rule 

Formula 

i 

2 

Velocity  in  Feet 
per  Minute   at  the 
Pitch  Diameter. 

Multiply  the  product  of 
the  diameter  in  inches  and 
the  number  of  revolutions 
per  minute  by  0.262. 

7=0.262  DR 

Allowable    Unit 
Stress    at     given 
Velocity. 

Multiply  the  allowable 
static   stress  by  600  and 
divide  the  result  by  the 
velocity  in  feet  per  min- 
ute plus  600. 

o        o    v       6o° 

SaX6oo  +  V 

3 

Maximum  Safe 
Tangential  Load  at 
Pitch  Diameter. 

Multiply    together    the 
allowable    stress   for   the 
given  velocity,  the  width 
of  face,  the  tooth  outline 
factor  and  the  difference 
between    the   pitch    cone 
radius  and  the  width  of 
face;   divide  the  result  by 
the  product  of  the  diam- 
etral pitch  and  the  pitch 
cone  radius. 

w     SFY(C-F) 

PC 

4 

Maximum  Safe 
Horsepower. 

Multiply  the  safe  load 
at  the  pitch  line  by  the 
velocity  in  feet  per  min- 
ute, and  divide  the  result 
by  33,000. 

HP-    WV 

33,000 

228  BEVEL   GEARING 

"Strength  and  Durability  of  Spur  Gearing,"  give  all  the  neces- 
sary information  for  calculating  the  strength  of  bevel  gears. 
The  formulas  and  factors  given  are  based  on  the  use  of  the  dfam- 
etral  pitch  of  the  gears,  and  constants  Y  given  in  the  factor  table 
are  figured  for  diametral  pitch.  If  the  circular  pitch  is  given, 
it  should  be  transformed  into  diametral  pitch  by  dividing  3.1416 
by  the  circular  pitch.  By  means  of  the  Formulas  (i)  to  (4),  the 
horsepower  which  can  be  transmitted  by  a  gear  of  given  pitch 
diameter  and  diametral  pitch,  running  at  a  given  number  of 
revolutions  per  minute,  can  be  found.  The  formulas  should  be 
used  in  the  order  given.  The  allowable  static  unit  stress  S8  is 
found  from  the  first  line  (velocity  =  o)  in  the  table  of  "  Working 
Stresses  used  in  the  Lewis  Formula  for  the  Strength  of  Gear 
Teeth,"  giverf  in  Chapter  III.  The  allowable  working  stress  S 
at  any  velocity  may  also  be  found  directly  from  the  table.  An 
example  showing  the  use  of  these  rules  and  formulas  is  given 
herewith. 

Calculate  the  maximum  load  at  the  pitch  line  which  can  be 
safely  allowed  for  the  bevel  gears  in  Fig.  i,  if  the  maximum 
allowable  static  stress  for  the  pinion  is  20,000  pounds,  and  for 
the  gear,  8,000  pounds  per  square  inch;  the  pinion  runs  at  300 
revolutions  per  minute.  The  calculations  for  the  pinion  are  as 
follows:  15 

N  =  c^i?  =  I5'5> approx- 

V  =  0.262  X  5  X  300  =  400  feet  per  minute  (about) .     .     .     (i) 

5  =  20,000  X =  12,000  pounds  per  square  inch .     (2) 

600  -}-  400 

w _  12,000x4x0.292x6.3 _ 286o pounds _  _  •• 1  .  (3)- 

3  X  10.3 

For  the  gear,  the  velocity  is  the  same  as  for  the  pinion.  The 
necessary  calculations  are  as  follows: 

.:;••'  ;|        N' = ^6° =  250> approx-  '',:'„',,  i  - 

5  =  8000  X — —  =  4800  pounds  per  square  inch  .     .     (2) 

600  +  400 

w  =  4800  X  4  X  0467  X  6.3 


STRENGTH  AND   DESIGN 


22<) 


10  M 

1 

^ 

* 

H 
? 

1 

2 

1 

0 

11 

0 

sg 

a 

Z 

* 

g 

i 

0 

a 

z   - 

UI     " 

[NUMBER  OF  TEETH 
CUTTING  ANGLE 

|  NO.  OF  CUTTER 

|  DIAMETRAL  PITCH 

|  SHAPE  OF  TEETH 

|  ADDENDUM 

|  WHOLE  DEPTH  OF  TO 

1  TOOTH  THICKNESS 

1  ADDENDUM  AT  SMALL 
TOOTH  THICKNESS  " 

230  BEVEL   GEARING 

The  gear  is,  therefore,  the  weaker  of  the  two,  and  thus  limits 
the  allowable  tooth  pressure.  The  maximum  horsepower  this 
gearing  will  transmit  safely  is  found  as  follows: 


H.P.   =  =  22  (4) 

33,000 

Durability  is  practically  of  as  much  importance  as  strength  in 
proportioning  bevel  gears,  but  unfortunately  no  information  is 
as  yet  available  for  making  satisfactory  comparisons  of  durability, 
so  that  the  usual  procedure  is  to  design  the  gears  for  strength 
alone,  assuming  then  that  they  will  not  wear  out  within  the  life- 
time of  the  machine  in  which  they  are  used. 

Simplified  Formulas  for  the  Strength  of  Bevel  Gears.  —  In 
Chapter  IV,  simplified  formulas  for  the  strength  of  gears  were 
given  by  means  of  which  the  circular  pitch  to  transmit  a  given 
horsepower  at  a  given  speed  could  be  found.  The  formulas 
relating  to  bevel  gears,  without  their  derivation,  are  repeated  in 
the  following  for  the  sake  of  completeness.  These  formulas  are 
based  on  the  assumption  that  the  pinion  (for  which  the  strength 
usually  is  calculated)  has  15  teeth.  If  the  number  of  teeth  in 
the  pinion  is  other  than  15,  multiply  the  horsepower,  as  given  in 
the  formulas  below,  by  (0.027  N  +  0.6),  in  which  N  =  number 
of  teeth.  In  the  formulas: 

P'  =  circular  pitch; 
H.P.  =  horsepower  to  be  transmitted; 
R.P.M.  =  revolutions  per  minute  of  the  pinion; 
D  —  pitch  diameter  of  15-  tooth  pinion. 


Cast-iron 
Cast-steel 

Bevel  Gear  P' 

Bevel  Gear  P' 
D 

Stress  according 
to  the  Lewis 
Tables 

Stress  %  of  that 
given  in  the 
Lewis  Tables 

*/S.oH.P.2 

°/«.oH.P.2 

V   R.P.M. 

V    R.P.M. 

'/o.SH.P.2 

'  Yi.SH.P.2 

V  R.p.M. 
=  4.77^'. 

V  R.P.M. 

The  formula  in  the  column  headed  "Stress  According  to  the 
Lewis  Tables"  is  used  for  ordinary  conditions  and  steady  drive. 


STRENGTH  AND   DESIGN 


231 


The  column  headed  "  Stress  Two- thirds  of  that  given  in  Lewis 
Tables"  should  be  used  for  gears  subjected  to  rapidly  varying 
loads  or  where  the  drive  is  often  started  and  stopped. 

Bearing  Pressures  Due  to  the  Action  of  Bevel  Gears  Under 
Load.  —  The  action  of  a  pair  of  bevel  gears  under  load  produces 
a  radial  pressure  on  the  bearings  due  to  the  tendency  of  the  driver 
to  climb  onto  the  driven  gear.  The  pressure  angle  of  the  in- 
volute teeth  also  produces  a  force  tending  to  push  the  gears  out 
of  engagement.  This  angular  thrust  may  be  resolved  into  a 
radial  bearing  pressure  and  a  direct  end  thrust  on  the  shaft. 


Machinery 


Fig.  2.  Gear  Dimensions  required 
to  determine  Radial  Pressure 
and  End  Thrust 


rig.  3.  Mean  Pitch  Diameter 
defined  for  Purposes  of  Cal- 
culation 


The  amount  of  radial  bearing  pressure  and  end  thrust  due  to  a 
given  load  on  a  given  gear  may  be  calculated  by  the  rules  and 
formulas  presented  in  the  following.  We  must  have  the  follow- 
ing data: 

1.  The  pitch  diameter,  width  of  face,  and  edge  angle  of  the 
gear;  the  edge  angle  is  the  same  as  the  pitch  cone  angle.    These 
dimensions  are  shown  in  Fig.  2.  , 

2.  The  horsepower  and  revolutions  per  minute,  or  the  torque 
in  inch-pounds  to  which  the  gear  will  be  subjected. 

3.  The  pressure  angle  of  the  teeth;  that  is,  whether  they  are 
14^-degree  standard  teeth,  or  2o-degree,  or  other  shape.    It 


232  BEVEL   GEARING 

makes  no  difference  whether  the  teeth  are  of  standard  length  or 
of  the  "stub-tooth"  variety. 

We  will  evidently  get  different  driving  pressures  on  the  teeth, 
depending  on  whether  we  use  the  pitch  diameter  at  the  large  end 
or  at  the  small  end  of  the  teeth  for  our  calculations.  We  should 
evidently  use  a  diameter  somewhere  between  these  two  to  get 
correct  results.  An  accurate  calculation  would  require  con- 
sideration of  the  elasticity  of  the  tooth,  which  would  be  some- 
what complicated.  It  will,  however,  be  accurate  enough  for 
practical  purposes  to  locate  our  mean  pitch  circle  at  one-third 
the  width  of  face  from  the  large  end  of  the  tooth.  The  diameter 
thus  located,  as  shown  in  Fig.  3,  may  be  considered  the  mean 
pitch  diameter  for  the  purposes  of  this  calculation.  This 
mean  pitch  diameter  may  be  obtained  either  by  measuring  an 
accurate  drawing  or  by  the  following  calculation: 

Rule  i.  To  find  the  mean  pitch  diameter,  multiply  two- thirds 
the  width  of  face  by  the  sine  of  the  edge  angle,  and  subtract  the 
product  from  the  pitch  diameter. 

For  the  tangential  tooth  pressure  we  have : 

Rule  2.  To  find  the  tangential  tooth  pressure  (which  equals 
the  direct  radial  pressure  on  the  bearing),  divide  the  torque  in 
inch-pounds  by  one-half  the  mean  pitch  diameter;  or, 

Rule  3.  Multiply  the  horsepower  transmitted  by  126,050,  and 
divide  by  the  product  of  the  revolutions  per  minute  and  the 
mean  pitch  diameter. 

The  next  step  is  to  find  the  thrust  due  to  the  tooth  pressure 
and  the  angularity  of  the  gear  in  the  direction  x  -  x  in  Fig.  4. 
This  may  be  done  by  the  regular  parallelogram  of  forces,  or  by 
calculation  as  follows: 

Rule  4.  To  find  the  angular  thrust  of  the  gear,  multiply  the 
tangential  tooth  pressure  by  the  tangent  of  the  pressure  angle  of 
the  tooth.  (Note:  For  a  i4§-degree  tooth  the  tangent  is  0.2586, 
for  a  i5-degree,  0.2680;  for  20  degrees,  0.3640;  for  22^  degrees, 
0.4142). 

This  angular  thrust,  as  shown  in  Fig.  4,  may  be  resolved  into 
two  components,  one  of  which  is  the  direct  thrust  on  the  shaft 
(which  may  be  taken  care  of  by  a  thrust  bearing),  and  the  other 


STRENGTH  AND  DESIGN 


233 


an  additional  radial  pressure  on  the  bearing.  These  quantities 
may  be  derived  by  the  parallelogram  of  forces  or  by  the  following 
calculation: 

Rule  5.  To  find  the  direct  thrust  on  the  shaft,  multiply  the 
angular  thrust  by  the  sine  of  the  edge  angle. 

Rule  6.  To  find  the  additional  bearing  pressure  due  to  the 
angular  thrust,  multiply  the  angular  thrust  by  the  cosine  of  the 
edge  angle. 


PRESSURE  ANGLE  OF  TOOTH 


Pr=TANGENTIAL  TOOTH  PRESSURE 
~~   DIRECT  RADIAL  PRESSURE  ON  BEARING 


Machinery 


Fig.  4.    Finding  Direct  Thrust  and  Additional  Radial  Bearing  Pressure 

It  is  evident  from  Fig.  5  that  the  bearing  pressure  due  to  the 
angularity  of  the  teeth  is  at  right  angles  to  that  due  to  the  tan- 
gential tooth  pressure  carrying  the  load.  The  resultant  total 
pressure  on  the  bearings  may  be  obtained  by  the  parallelogram 
of  forces,  or  by  calculation  as  follows: 

Rule  7.  To  find  the  total  radial  pressure  on  the  bearings,  add 
together  the  squares  of  the  tangential  tooth  pressure  and  the 
additional  radial  thrust  on  the  bearing  due  to  the  angular  thrust, 
and  extract  the  square  root  of  the  sum. 


234  BEVEL  GEARING 

Formulas  Embodying  the  Rules  Given.  —  The  foregoing  rules 
may  be  expressed  in  the  following  formulas: 

Dm  =  J9-(|FXsina)  .........     (i) 


_  i26,Q5oH.P. 


T0  =  PrX  tan/3    .....  ••/,*    .    .    .    .     (4) 

Td  =  TaXsina    ....    .    ...    .    .    .     (5) 

Pt  =  Ta  X  cos  a     .     .  ^     .     .     .     .     .     .     .     .     (6) 


Pb  =  VPS  +  P?    ...........     (7) 

in  which, 

D  =  pitch  diameter; 

F  =  width  of  face; 

a  =  edge  angle  (same  as  pitch  cone  angle)  ; 
Dm  =  mean  pitch  diameter; 
M  =  torque  or  turning  moment  in  inch-pounds; 

R  =  revolutions  per  minute; 

Pr  =  direct  radial  pressure  on  bearing  =  tangential  tooth 
pressure; 

/3  =  pressure  angle  of  involute  tooth; 
Ta  =  angular  thrust  of  gears; 

Td  -  direct  thrust  on  shaft  due  to  angular  thrust  of  gears; 
Pt  =  additional  radial  pressure  on  bearings  due  to  angular 

thrust; 

pb  =  total  radial  pressure  on  bearing; 
H.P.  =  horsepower  transmitted. 

Example  of  Calculation  of  Thrust.  —  As  an  example,  take  the 
case  of  the  gear  shown  in  Figs.  4  and  5,  the  dimensions  of  which 
are  as  follows:  Pitch  diameter  =  6  inches;  face  =  if  inch;  edge 
angle  =  63  degrees  26  minutes;  horsepower  =  19,  or  torque  = 
1500  inch-pounds  at  800  R.P.M.;  pressure  angle  of  tooth  =  20 
degrees. 

The  radial  pressure  and  thrust  on  the  bearing  are  found  by 
calculation  from  the  given  rules  and  formulas,  as  follows. 


STRENGTH  AND   DESIGN 


235 


Graphical  calculations  giving  the  same  results  by  the  parallelo- 
gram of  forces  are  indicated  in  Figs.  4  and  5. 

Dn  =  6  -  |  X  0.894  =  5.330  inches  .....     (i) 
pr  =    300Q    =  560  pounds,  about  .....     (2) 


56°  pounds,  about. 


Ta  —  560  X  0.364  =  204  pounds 
Td  =  204  X  0.894  =  182  pounds. 
Pt  =  204  X  0.447  =  91  pounds 
Pb  =  V56o2  +  9i2  =  567  pounds. 


(3) 

(4) 
(5) 
(6) 
(7) 


J=,==TANGENTIAL  TOOTH  PRESSURE 
AT  MEAN  DIAMETER 


Machinery 


Fig.  5.    Finding  the  Total  Radial  Pressure  on  the  Bearing 

Effect  of  Pressure  Angle.  —  It  will  be  seen  from  (7)  that  the 
pressure  angle  of  the  tooth  has  a  practically  negligible  effect  in 
increasing  the  radial  bearing  pressure.  In  the  case  of  spur  gears, 
this  effect  is  so  small  that  the  angular  pressure  need  not  be  reck- 
oned with,  and  in  bevel  gears  the  effect  is  even  smaller,  since  a 
good  share  of  the  angular  pressure  is  transmitted  into  thrust. 
It  is,  therefore,  not  necessary  in  practice  to  carry  the  calculations 
beyond  the  fifth  rule  or  formula. 

Total  Bearing  Pressure.  —  The  total  radial  bearing  pressure 
thus  found  is,  of  course,  distributed  between  the  bearings  of  the 


236 


BEVEL   GEARING 


shaft,  usually  two  in  number,  in  accordance  with  the  principle 
of  moments.  (See  MACHINERY'S  Handbook,  pages  338  and  339.) 
The  pressure  on  each  bearing  resulting  from  the  gear  action,  as 
calculated  above,  must  be  combined  by  the  parallelogram  of 
forces  in  the  same  way  as  shown  in  Fig.  5  with  any  other  pressures 
arising  in  these  bearings  from  other  gears,  belt  pulleys,  brakes  or 
similar  loads  on  the  same  shaft. 

Design  of  Bevel  Gearing.  —  So  far  we  have  dealt  with  design 
as  relating  to  calculations.     In  the  following  will  be  discussed 


Machinery 


Fig.  6.    T-arm  Style  of  Bevel  Gear  for  Heavy  Work 

the  application  of  the  calculated  dimensions,  the  determination 
of  the  factors  left  to  the  judgment,  and  the  recording  of  the  design 
in  the  drawing. 

Bevel  Gear  Blanks.  —  Various  forms  may  be  given  to  the 
blanks  or  wheels  on  which  bevel  gear  teeth  are  cut,  depending  on 
the  size,  material,  service,  etc.,  to  be  provided  for.  The  pinion 
type  of  blank  is  shown  in  Fig.  12,  Chapter  X,  and  is  used  mostly, 
as  indicated  by  the  name,  for  gears  of  a  small  number  of  teeth 
and  small  pitch  cone  angle.  Where  the  diameter  of  the  bore 


STRENGTH  AND   DESIGN  237 

comes  too  near  to  the  bottoms  of  the  teeth  at  the  small  end,  it  is 
customary  to  omit  the  recess  indicated  by  dimension  z,  and  leave 
the  front  face  of  the  pinion  blank  as  in  the  case  shown  in  Fig.  i 
of  the  present  chapter. 

For  gears  of  a  larger  number  of  teeth,  the  web  type  shown  in 
Fig.  9,  Chapter  X,  is  appropriate.  This  does  not  require  to  be 
finished  all  over,  as  the  sides  of  the  web,  the  outside  diameter 
of  the  hub,  and  the  under  side  of  the  rim  may  be  left  rough 
if  desired. 

A  steel  gear  suitable  for  very  heavy  work  is  shown  in  Fig.  6 
of  the  present  chapter.  Here  the  web  is  reinforced  by  ribs. 
The  web  may  be  cut  out  so  that  the  rim  is  supported  by  T-shaped 
arms,  as  shown.  This  makes  a  very  stiff  wheel  and  at  the  same 
time  a  very  light  one,  when  its  strength  is  considered.  Where 
the  pitch  cone  angle  is  so  great  that  the  strengthening  rib  would 
be  rather  narrow  at  the  flange,  it  may  be  given  the  form  shown 
in  Fig.  i  in  place  of  that  shown  in  Fig.  6. 

General  Considerations  Relating  to  Design.  —  The  perform- 
ance of  the  most  carefully  designed  and  made  bevel  gears  depends 
to  a  considerable  extent  on  the  design  of  the  machine  in  which 
they  are  used.  When  the  shafts  on  which  a  pair  of  bevel  gears 
are  mounted  are  poorly  supported  or  poorly  fitted  in  their  bear- 
ings, the  pressure  of  the  driving  gear  on  the  driven  causes  it  to 
climb  up  on  the  latter,  throwing  the  shafts  out  of  alignment. 
This  in  turn  causes  the  teeth  to  bear  with  a  greater  pressure  at 
one  end  of  the  face  (usually  on  the  outer  end)  than  the  other, 
thus  making  the  tooth  more  liable  to  break  than  is  the  case  where 
the  pressure  is  more  evenly  distributed.  It  is  important,  there- 
fore, to  provide  rigid  shafts  and  bearings  and  careful  workman- 
ship for  bevel  gearing. 

The  question  of  alignment  of  the  shafts  should  be  considered 
in  deciding  on  the  width  of  face  of  the  gear.  Making  the  width 
of  the  face  more  than  one-third  of  the  pitch  cone  radius  adds 
practically  nothing  to  the  strength  of  the  gear  even  theoretically, 
since  the  added  portion  is  progressively  weaker  as  the  tooth  is 
lengthened,  as  has  been  explained.  In  addition  to  this,  there  is 
the  danger  that  through  springing  of  the  shafts  or  poor  work- 


238  BEVEL   GEARING 

manship,  the  load  will  be  thrown  onto  the  weak  end  of  the  tooth, 
thus  fracturing  it.  For  this  reason  it  may  be  laid  down  as  a 
definite  rule  that  there  is  nothing  to  be  gained  by  making  the 
face  of  the  bevel  gear  more  than  one-third  of  the  pitch  cone 
radius. 

The  Brown  &  Sharpe  gear  book  gives  a  rule  for  the  maximum 
width  of  face  allowable  for  a  given  pitch.  The  width  of  face 
should  not  exceed  i\  times  the  circular  pitch  or  8  divided  by 
the  diametral  pitch.  This  rule  is  also  rational  since  the  danger 
to  the  teeth  from  the  misalignment  of  the  shaft  increases  both 
with  the  width  of  face  and  with  the  decrease  of  the  size  of  the 
tooth,  so  that  both  of  these  should  be  reckoned  with.  In  de- 
signing gearing  it  is  well  to  check  the  width  of  face  from  the  rule 
relating  to  the  pitch  cone  radius  and  that  relating  to  the  pitch 
as  well,  to  see  that  it  does  not  exceed  the  maximum  allowed 
by  either. 

Model  Bevel  Gear  Drawing.  —  It  is  not  enough  for  the  de- 
signer to  carefully  calculate  the  dimensions  of  a  set  of  bevel  gear- 
ing. In  addition  to  this  he  has  the  important  task  of  recording 
these  dimensions  in  such  a  form  that  they  will  be  intelligible  to  an 
intelligent  workman,  and  will  plainly  furnish  him  every  point  of 
information  needed  for  the  successful  completion  of  the  work 
without  further  calculation.  A  drawing  which  practically  fills 
these  requirements  is  shown  in  Fig.  i.  The  arrangement  of 
this  drawing  and  the  amount  and  kind  of  information  shown 
on  it  are  based  on  the  drafting-room  practice  of  the  Brown  & 
Sharpe  Mfg.  Co.,  with  a  few  slight  changes. 

In  general,  the  dimensions  necessary  for  turning  the  blank  have 
been  given  on  the  drawing  itself,  while  those  for  cutting  the  teeth 
are  given  in  tabular  form.  All  the  dimensions  were  calculated 
from  Rules  (i)  to  (21)  in  Chapter  X,  and  may  be  checked  for 
practice  by  the  reader.  It  will  be  noticed  that  limits  are  given 
for  the  important  dimensions.  This  should  always  be  done  for 
manufacturing  work  which  is  inspected  in  its  course  through  the 
shop.  It  ought  to  be  done  even  when  a  single  gear  is  made,  as  it 
is  exceedingly  difficult  to  properly  set  a  gear  if  the  workman  does 
not  work  close  enough.  There  is  no  sense,  however,  in  asking 


STRENGTH  AND   DESIGN  239 

him  to  work  to  thousandths  of  an  inch  on  blanks  like  these,  so 
he  should  be  given  some  notion  as  to  the  accuracy  required  by 
limits  such  as  shown. 

It  is  assumed  that  the  gears  are  to  be  cut  with  rotary  cutters. 
It  is  unusual  to  do  this  with  a  pitch  as  coarse  as  this,  though  there 
are  machines  on  the  market  capable  of  handling  such  work.  In 
gear-cutting  machines  using  form  cutters,  the  blanks  are  located 
for  axial  position  by  the  rear  face  of  the  hub.  It  is  necessary 
also  to  leave  stock  at  this  place  for  fitting  the  gears  in  the  ma- 
chine. It  will  be  seen  that  the  dimension  for  bevel  gears  and 
pinions  from  the  outside  edge  of  the  blank  to  the  rear  face  of  the 
hub  is  marked  "Make  all  alike."  This  means  that  the  same 
amount  of  stock  should  be  left  on  all  the  gears  in  a  given  lot  so 
that  after  the  machine  is  set  for  one  of  them,  it  will  not  be  neces- 
sary to  alter  the  adjustment  for  the  remainder. 

There  are  one  or  two  dimensions  which  are  not  given  directly 
by  Rules  (i)  to  (21).  One  of  these  is  the  distance  4.57  inches 
from  the  outside  edge  of  the  teeth  to  the  finished  rear  face  of  the 
hub  of  the  gear.  This  dimension  is  commonly  scaled  from  an 
accurate  drawing,  but  it  may  be  calculated  by  subtracting  the 
vertex  distance  from  the  distance  between  the  pitch  cone  vertex 
0  and  the  rear  face  of  the  hub.  This  gives  6f  —  2.1764  equals 
4.57  inches  (about)  as  dimensioned.  Another  dimension  not 
directly  calculated  is  the  over-all  length  of  the  pinion.  This 
may  be  obtained  by  subtracting  the  vertex  distance  at  the  small 
end  (marked  j  in  illustrations  with  rules  and  formulas  in  Chapter 
X)  from  the  distance  between  the  vertex  and  the  rear  face  of  the 
hub,  giving  4.93  inches  as  shown. 

In  the  tabular  dimensions  for  cutting  the  teeth,  most  of  the 
figures  are  self-explanatory.  The  fact  that  in  this  particular 
case  a  2o-degree  form  of  tooth  has  been  adopted  to  avoid  the 
undercut  in  small  pinions  (see  Chapter  X,  section  on  "Systems 
of  Tooth  Outlines  used  for  Bevel  Gearing")  is  indicated  in  the 
table. 

The  number  of  cutter  is  selected  from  the  table  "Formed  Cut- 
ters for  Involute  Bevel  Gear  Teeth,"  Chapter  X,  in  accordance 
with  the  number  of  teeth  (N')  in  equivalent  spur  gear,  as  deter- 


240 


BEVEL  GEARING 


mined  by  Rule  (20)  in  the  formulas  in  the  same  chapter.  This 
is  15.4  for  the  pinion,  and  247  for  the  gear,  giving  a  No.  i  and 
No.  7  cutter  respectively.  These  cutters  are  marked  special, 
owing  to  the  fact  that  they  are  20  degrees  involute  instead  of  14^ 
degrees.  They  would  be  special  under  any  circumstances,  how- 
ever, since  the  width  of  face  for  these  gears  (4  inches)  is  more 
than  |  the  pitch  cone  radius,  which  figures  out  to  10.3097  inches. 
Standard  bevel  gear  cutters  are  only  made  thin  enough  to  pass 
through  the  teeth  at  the  small  end  when  the  width  of  face  is  not 


•MAKE  ALL  ALIKE 


Figs.  7  and  8.    Additional  Dimensions  for  Gears  to  be  cut  by  the 
Templet  Planing  Process  or  the  Gleason  Generating  Machine 

more  than  J  the  pitch  cone  radius.     For  this  reason  cutters 
thinner  than  the  standard  would  have  to  be  used. 

In  bevel  pinions  of  the  usual  form,  such  as  shown  in  Fig.  12, 
Chapter  X,  dimension  z  there  given  has  to  be  furnished.  This 
may  be  scaled  from  a  carefully  made  drawing,  or  may  be  calcu- 
lated by  subtracting  the  length  of  the  bore  of  the  pinion  from  the 
over-all  length,  the  latter  being  obtained  as  described  for  the 
pinion  in  Fig.  i.  Such  dimensions  do  not  need  to  be  given  in 
thousandths  of  an  inch  on  moderately  large  work.  It  is  also 
not  necessary  to  give  the  angles  any  closer  than  the  quarter 
degree,  as  few  machines  are  furnished  with  graduations  which 


STRENGTH  AND  DESIGN 


241 


can  be  read  finer  than  this.  In  order  to  check  the  calculations 
carefully,  however,  it  is  wise,  as  previously  described,  to  make 
them  with  considerable  accuracy,  using  tables  of  sines  and  tan- 
gents which  read  to  five  figures.  After  the  dimensions  are  calcu- 
lated, they  may  be  put  in  more  approximate  form  for  the  drawing. 
The  gear  drawing  in  Fig.  i  is  dimensioned  more  fully,  perhaps, 
than  is  customary,  especially  in  shops  having  a  large  gear-cutting 
department,  where  the  foreman  and  operators  are  experienced 
and  have  access  to  tables  and  records  of  data  for  bevel  gear  cut- 


MAKE  ALL  ALIKE 


Machinery 


Figs.  9  and  10.    Additional  Dimensions  for  Gears  to  be  cut  on  the 
Bilgram  Generating  Machine 

ting.     Every  dimension  given  is  useful,  however,  and  it  is  a  good 
plan  to  include  them  all,  especially  on  large  work. 

Dimensioning  Drawings  for  Gears  with  Planed  Teeth.  —  The 
machine  on  which  the  teeth  of  a  gear  are  to  be  cut  determines 
to  some  extent  the  dimensions  which  the  workman  needs,  so  this 
should  be  taken  into  account  in  making  the  drawing.  For  gears 
which  are  to  be  cut  on  a  templet  planing  machine,  the  dimensions 
given  in  Fig.  i  may  be  followed  in  general.  Further  dimensions 
are  needed,  however,  to  set  the  blank  so  that  the  vertex  of  the 
pitch  cone  corresponds  with  the  central  axis  of  the  machine.  For 


242  BEVEL  GEARING 

gears  with  pitch  cone  angle  greater  than  45  degrees,  this  may  be 
obtained  from  dimension  X,  as  given  in  Fig.  7,  or,  better,  from 
dimension  /.  For  gears  smaller  than  45  degrees,  C  (Fig.  8)  may 
be  given. 

There  are  two  commercial  forms  of  gear  generating  machines  in 
general  use  in  this  country  for  planing  the  teeth  of  bevel  gears. 
These  are  the  Gleason  and  Bilgram  machines.  Since  the  methods 
of  supporting  the  gears  are  different,  the  drawings  should  be 
dimensioned  to  suit,  if  it  is  known  beforehand  how  they  are  to  be 
cut.  For  the  Gleason  machine  the  dimensioning  shown  in  Figs. 
7  and  8  should  be  given,  in  addition  to  that  shown  in  Fig.  i. 
The  angles  a  and  <j>,  the  pitch  cone  angle  and  dedendum  angle, 
respectively,  may  well  be  put  in  the  table  of  dimensions  instead 
of  on  the  drawing.  The  distance  from  the  outside  corner  of  the 
teeth  to  the  rear  face  of  the  hub  should  be  made  alike  for  all 
similar  gears  in  the  lot,  the  same  as  for  gears  which  are  to  be  cut 
by  the  form  cutters  or  the  templet  process.  The  cutting  angle 
may  be  omitted  from  the  drawing. 

The  method  of  dimensioning  for  the  Bilgram  gear  planer  is 
shown  in  Figs.  9  and  10.  Angles  a  and  $  should  be  given  in  the 
table  as  before.  Dimension  5  is  used  for  setting  on  gears  of 
large  pitch  cone  angle,  and  dimension  C  or  the  pitch  cone  radius 
for  those  of  small  pitch  cone  angle  (less  than  45  degrees).  It  is 
a  good  idea  to  give  both  of  these  dimensions  for  both  gear  and 
pinion,  so  that  the  setting  may  be  checked  by  two  different 
methods.  In  this  machine  the  dimension  to  be  marked  "Make 
all  alike"  should  be  given  as  shown. 


CHAPTER  XII 

METHODS   FOR   FORMING   THE    TEETH    OF 
BEVEL   GEARS 

IT  will  be  impossible  in  the  comparatively  short  space  we  are 
able  to  devote  to  the  subject  of  forming  the  teeth  in  bevel  gears 
to  do  more  than  give  the  bare  outlines  of  the  ingenious  mechan- 
isms which  have  been  devised  for  this  work.  Almost  any  one 
of  the  methods  to  be  described  would  require  several  pages  and 
many  illustrations  to  explain  the  details  of  its  application,  and 
special  books  devoted  to  this  subject  alone  are  available.  (See 
"Gear  Cutting  Machinery,"  by  R.  E.  Flanders.)  We  can,  how- 
ever, in  the  comparatively  short  descriptions  here  given,  get  an 
understanding  of  the  principles  of  operation  of  each  method. 

Comparison  between  Spur  and  Bevel  Gear  Cutting.  —  The 
changes  required  in  spur  gear  cutting  devices  to  adapt  them 
for  cutting  bevel  gears,  made  necessary  by  the  difference  in  the 
nature  of  the  two  forms  of  gearing,  are  illustrated  in  Figs,  i,  2 
and  3.  The  action  of  a  pair  of  mating  spur  gears  may  be  seen 
and  studied  on  the  plane  perpendicular  to  their  axes.  To  be 
understood  correctly,  the  action  of  bevel  gearing  must  be  ob- 
served on  a  spherical  surface.  In  Fig.  i  are  shown  three  bevel 
gears  with  axes  OA,  OB  and  OC.  The  bevel  gear  on  axis  OA  is 
of  the  form  known  as  the  "crown  gear."  It  is  practically  a  rack 
bent  in  a  circle  about  center  0.  Pinion  OB  and  gear  OC  are 
familiar  types  of  bevel  gears.  In  Fig.  2  are  shown  the  pitch  sur- 
faces of  the  gears  in  the  preceding  figure.  It  will  be  seen  in  that 
figure  that  the  pitch  lines  of  the  gear  on  the  axis  OC,  for  instance, 
converge  at  the  center  O.  These  pitch  lines  represent  a  conical 
pitch  surface  which  is  shown  cut  out  from  a  sphere  on  axis  OC 
in  Fig.  2.  In  a  similar  way  the  cone  about  axis  OB  represents 
the  pitch  surface  of  the  pinion  in  Fig.  i,  while  the  plane  face  of 
the  hemisphere  at  the  left  in  Fig.  2  is  the  pitch  surface  of  the 

243 


244 


BEVEL  GEARING 


crown  gear  of  the  preceding  figure.  If  we  wish  to  draw  accurate 
representations  of  the  teeth  of  the  bevel  gears  in  Fig.  i,  in  order 
to  study  their  action  in  the  same  way  that  we  can  when  drawing 
the  teeth  of  spur  gears  on  the  plane  surface  of  the  drawing-board, 
we  would  have  to  draw  them  on  surfaces  of  the  sphere  from  which 
the  pitch  cones  in  Fig.  2  are  cut.  The  pitch  circles,  etc.,  of  the 
various  gears  would  be  struck  from  centers  located  at  the  points 
where  the  various  axes  OA ,  OB  and  OC  break  through  the  surface 
of  the  sphere.  Except  for  the  different  surfaces  on  which  the 
drawing  would  be  done,  the  procedure  would  be  identical  with 
that  for  spur  gears.  It  should  be  noted  that  straight  lines  on 
spherical  surfaces  are  represented  by  great  circles  —  that  is  to 
say,  by  the  intersection  with  the  surface  of  planes  passing  through 
the  center  of  the  sphere. 


Pigr.  i. 


Fig.  2. 


Fig.  3.     Machinery 


Figs,  i,  2  and  3.  Illustrating  the  Spherical  Basis  of  the  Bevel  Gear  and 
Tredgold's  Approximation  for  developing  the  Outlines  of  the  Teeth  on  a 
Plane  Surface 

Tredgold's  Approximation.  —  Owing  to  the  impracticability  of 
the  sphere  as  a  drawing-board,  a  process  known  as  "Tredgold's" 
is  usually  followed  for  approximately  laying  out  the  teeth  of  bevel 
gears.  This  is  shown  in  Fig.  3  applied  to  the  same  case  as  in  the 
two  preceding  figures.  The  conical  pitch  surfaces  vanishing  at 
the  center  0  are  identical  with  those  in  Fig.  2,  as  is  also  the  plane 
circular  face  of  the  crown  gear.  For  the  bevel  gear  and  pinion, 
however,  the  teeth  are  supposed  to  be  drawn  and  the  action 
studied  on  surfaces  of  cones  complementary  to  the  pitch  cones  — 
that  is,  on  the  cones  with  apexes  at  c  and  b.  The  surface  of  these 
cones  can  be  developed  on  a  flat  piece  of  paper,  as  shown  for  that 
on  axis  OC  in  Fig.  3,  in  which  case  the  pitch  line  becomes  xy,  as 


FORMING  THE  TEETH 


245 


illustrated.  Teeth  drawn  on  this  pitch  line,  as  for  a  spur  gear, 
may  be  laid  out  on  the  conical  surface  and  used  as  the  outlines  of 
bevel  gear  teeth.  The  difference  in  the  shape  of  tooth  obtained 
under  the  same  system  as  the  two  methods  shown  in  Figs.  2  and 
3,  is  so  slight  as  to  be  negligible,  except  perhaps,  in  gears  having 
very  few  teeth.  Whatever  the  method  pursued  for  laying  out 
or  studying  the  action,  all  the  elements  of  which  the  teeth  are 
formed  consist  of  straight  lines  which  meet  at  the  center  O  of  the 
pitch  cones;  consequently  the  teeth  grow  small  toward  the  inner 
end,  vanishing  at  the  center  if  they  are  carried  that  far. 


Machinery 


Fig.  4.    Shaping  the  Teeth  of  a  Bevel  Gear  by  the  Formed  Cutter  Process 

Five  Principles  of  Action.  —  All  of  the  five  principles  of  action 
on  which  spur  gear  teeth  may  be  formed  (the  formed  tool,  the 
templet,  the  odontographic,  the  describing-generating  and  the 
molding-generating  principles)  may  be  also  applied  to  the  cutting 
of  bevel  gears,  although  the  describing-generating  principle  has 
never  been  so  used. 

The  Formed  Tool  Principle.  —  The  application  of  this  principle 
is  illustrated  in  Fig.  4,  where  a  form  cutter  is  shown  shaping  one 
side  of  the  tooth  of  a  bevel  gear.  The  gear  blank  is  tipped  up  to 
cutting  angle  a  and  fed  beneath  the  cutter  in  the  direction  of  the 
arrow.  It  will  be  immediately  seen  from  an  examination  of  the 


246  BEVEL  GEARING 

figure  that  the  form  tool  process  is  by  necessity  approximate.  It 
is  evident  that  the  right-hand  side  of  the  cutter  is  reproducing 
its  own  unchanging  outline  along  the  whole  length  of  the  face  of 
the  tooth  at  the  right.  This  form  should  not  be  unchanging  for, 
as  has  been  explained,  the  teeth  and  the  space  between  them 
grow  smaller  towards  the  apex  of  the  pitch  cone,  where  they 
finally  vanish,  so  it  is  evident  that  the  outline  of  a  tooth  at  the 
small  end  should  be  the  same  as  that  at  the  large  end,  but  on  a 
smaller  scale,  and  not  a  portion  of  the  exact  outline  at  the  large 
end,  as  produced  by  the  formed  tool  process  and  as  shown  in  the 
figure.  To  make  this  error  as  small  as  possible,  it  is  customary 
to  use  a  cutter  which  gives  the  proper  shape  at  the  large  end, 


SECTION  OF  QEAR  BEING  CUT 


Machinery 


Fig.  5.    The  Templet  Principle  for  Forming  the  Teeth  of  Bevel  Gears 

and  set  the  blank  so  that  the  tooth  is  cut  to  the  proper  thick- 
ness at  the  small  end.  This  leaves  the  top  of  the  tooth  at  the 
small  end  too  thick,  an  error  which  is  often  remedied  by  filing. 
Of  course,  the  principle  is  the  same  with  the  formed  planer  or 
shaper  tool  as  with  the  formed  milling  cutter,  and  the  errors 
involved  in  the  process  are  also  identical.  It  is  evident  that 
but  one  side  of  the  tooth  space  can  be  cut  at  a  time,  so  that  at 
least  two  cuts  around  will  have  to  be  taken.  The  method,  as 
applied  in  practice,  will  be  described  in  detail  in  the  next  chapter. 
The  Templet  Principle.  —  This  principle  is  illustrated  in  Fig.  5, 
in  skeleton  form  only.  A  former  is  used  which  has  the  same  out- 
line as  would  the  tooth  of  the  gear  being  cut,  if  the  latter  were 
extended  as  far  from  the  apex  of  the  pitch  cone  as  the  position 


FORMING  THE  TEETH 


247 


in  which  the  former  is  placed.  The  tool  is  carried  by  a  slide 
which  reciprocates  it  back  and  forth  along  the  length  of  the  tooth 
in  a  line  of  direction  (OX,  OY,  etc.),  which  passes  through  the 
apex  0  of  the  pitch  cone.  This  slide  may  be  swiveled  in  any 
direction  and  in  any  plane  about  this  apex,  and  its  outer  end  is 
supported  by  the  roller  on  the  former.  With  this  arrangement, 
in  the  case  shown,  as  the  slide  is  swiveled  inward  about  the  apex, 
the  roll  runs  up  on  the  former,  raising  the  slide  and  the  tool  so  as 
to  reproduce  on  the  proper  scale  the  outline  of  the  former  on 
the  tooth  being  cut.  Since  the  movement  of  the  tool  is  always 
toward  the  apex  of  the  pitch  cone,  the  tooth  elements  vanish 


SECTION  OF  GEAR  BEING  CUT 


.CONICAL  LINK  WHICH  GUIDES 
THE  TOOL  SLIDE 


Machinery 


Fig.  6.    The  Odontographic  Principle  for  Forming  the  Teeth  of  Bevel  Gears 

at  this  point,  and  the  outlines  are  similar  at  all  sections  of  the 
tooth,  although  with  a  gradually  decreasing  scale  as  the  apex  is 
approached  —  all  as  required  for  correct  bevel  gearing. 

The  arrangement  thus  shown  diagrammatically  is  modified  in 
various  ways  in  different  machines,  but  the  movement  imparted 
to  the  tool  in  relation  to  the  work  is  the  same  in  all  cases  where 
the  templet  principle  is  employed,  no  matter  what  the  connection 
between  the  former  and  the  tool  may  be. 

The  Odontographic  Principle.  —  As  explained  for  spur  gears 
in  Fig.  3,  Chapter  VIII,  it  is  often  possible  to  approximate  the 
exact  curves  required  for  the  teeth  of  gears  by  mechanisms  which 
make  use  of  circular  arcs  or  other  easily  generated  curves.  In 


248 


BEVEL   GEARING 


Fig.  6,  of  the  present  chapter,  is  shown  in  diagrammatic  form  an 
arrangement  for  obtaining,  by  means  of  link  work,  a  close  approxi- 
mation to  the  exact  form  of  an  involute  outline,  such  as  might  be 
produced  by  the  templet  in  Fig.  5,  for  instance.  This  true  invo- 
lute outline  may  be  very  closely  approximated  by  a  circle  drawn 
on  the  surface  of  a  sphere.  To  give  this  required  circular  move- 
ment to  the  point  of  the  tool,  the  slide  on  which  the  tool  recipro- 
cates may  be  constrained  by  a  link  as  shown,  pivoted  at  the  base 
to  the  frame  of  the  machine,  and  at  the  upper  end  to  the  slide. 
The  axes  of  these  pivots  should  pass  through  the  apex  of  the  pitch 
cone,  as  required  by  the  spherical  nature  of  the  bevel  gear.  This 


Machinery 


Fig.  7.    The  Impression  Operation  applied  to  Forming  the  Teeth  of  a 
Bevel  Gear  by  the  Molding-generating  Process 

link  work  (which  is  thus  of  the  "conical"  type)  if  properly  pro- 
portioned and  located,  will  guide  the  tool  slide  and  the  tool  point 
in  very  nearly  the  same  way  as  a  properly  constructed  templet, 
used  as  shown  in  Fig.  5. 

The  Molding-generating  Principle.  —  The  counterpart  of  the 
spur  gear  process  shown  in  Fig.  5,  Chapter  VIII,  is  illustrated  for 
the  bevel  gears  in  Fig.  7.  Here  a  correctly  formed  gear  is  being 
rotated  in  the  proper  position  and  in  the  proper  ratio  with  a 
plastic  blank.  This  operation,  as  in  the  case  of  the  spur  gear, 
forms  teeth  in  the  plastic  blank  which  are  properly  shaped  to 
mesh  with  the  forming  gear  or  with  any  other  gear  of  the  same 
series.  Fig.  6,  Chapter  VIII,  has  no  possible  counterpart  in  the 
cutting  of  bevel  gears. 


FORMING  THE   TEETH 


249 


To  fully  understand  the  principle  outlined  in  the  previous  para- 
graph, suppose  we  have  a  bevel  gear  blank  made  of  some  plastic 
material,  such  as  clay  or  putty.  By  transposing  Formula  (34), 


Chapter  X,  sin  ap  = 


N 


to  read  N0  = 


N 


,  it  is  evidently  possible 


to  make  a  crown  gear  which  will  mesh  properly  with  any  bevel 
gear,  such  as  the  one  we  wish  to  form.  If  this  crown  gear  and 
the  plastic  blank  are  properly  mounted  with  relation  to  each 
other  and  rolled  together,  the  tooth  of  the  crown  gear  will  form 
tooth  spaces  and  teeth  of  the  proper  shape  in  the  blank.  This 
is  the  foundation  principle  of  the  molding-generating  method. 


MASTER  GEAR 


RECIPROCATING  TOOL  SLIDE 


Machinery 


Fig.  8.    The  Shaping  or  Planing  Operation  applied  to  the  Molding- 
generating  Principle  of  Forming  Teeth  of  Bevel  Gears 

In  practice  we  have  blanks  of  solid  steel  or  iron  to  machine 
instead  of  those  made  from  putty  or  clay,  so  the  operation  has  to 
be  modified  accordingly.  Fig.  8  shows  in  diagrammatic  form  an 
apparatus  for  using  the  shaping  or  planing  operation  with  the 
molding-generating  principle.  Here  the  crown  gear  is  of  larger 
diameter  than  is  required  to  mesh  with  the  gear  being  cut,  and 
it  engages  a  master  gear  keyed  to  the  same  shaft  as  the  gear  being 
cut,  and  formed  on  the  same  pitch  cone.  If  the  teeth  of  the 
crown  gear,  instead  of  being  comparatively  narrow  as  shown, 
were  extended  clear  to  the  vertex  0,  they  would  mesh  properly 
with  the  gear  to  be  cut.  The  tooth,  as  shown,  has  a  line  of 
movement  such  that  the  point  of  the  tooth  travels  along  line  OX, 
which  is  the  corner  of  a  tooth  of  an  imaginary  extension  of  the 


250  BEVEL   GEARING 

crown  gear.  This  crown  gear  has  a  plane  face  (see  reference  to 
"octoid"  form  of  tooth,  Chapter  X),  and  the  cutting  edge  of  the 
tooth  is  straight  and  set  to  mesh  the  face  of  the  tooth.  As  it 
is  reciprocated  by  suitable  mechanism  (not  shown)  the  cutting 
edge  represents  a  face  of  the  imaginary  crown  gear  tooth.  If 
now  the  master  gear  and  crown  gear  are  rolled  together  and  the 
reciprocating  tool  starts  in  at  one  side  of  the  gear  to  be  cut  and 
passes  out  at  the  other,  the  straight  cutting  edge  of  the  tooth 
will  generate  one  side  of  a  tooth  in  the  gear  to  be  cut  in  the  same 
way  as  if  the  extended  tooth  of  the  crown  gear  were  rolling  its 
shape  on  one  side  of  the  tooth  of  a  plastic  blank.  This  simple 
mechanism  has,  of  course,  to  be  complicated  by  provisions  for 
cutting  both  sides  of  the  tooth,  and  for  indexing  the  work  from 
one  tooth  to  the  other  so  as  to  complete  the  entire  gear.  Ar- 
rangements have  to  be  made  also  to  make  the'machine  adjustable 
for  bevel  gears  of  all  angles,  numbers  of  teeth  and  diameters 
within  its  range. 

The  use  of  the  three  principles  illustrated  in  Figs.  4,  5  and  8 
is  not  limited  to  the  cutting  operation  shown  for  each  case.  In 
Fig.  4,  for  instance,  a  formed  planer  or  shaper  tool  may  be  used 
as  well  as  a  formed  milling  cutter.  Templet  machines  have  been 
made  in  which  a  milling  cutter  is  used  instead  of  a  shaper  tool. 
This  is  true  also  of  the  molding-generating  principle  shown  in 
Fig.  8. 

Four  Methods  of  Operation  —  By  Impression.  —  The  same 
four  methods  of  operation  as  for  spur  gears  may  be  applied  to 
the  molding-generating  principle,  and  quite  generally  to  the  other 
principles  as  well.  Instead  of  using  for  illustration  a  rack  as  the 
generating  member,  we  will  have  to  use  its  bevel  gear  counter- 
part, the  crown  gear  shown  in  Fig.  i.  The  impression  method 
would  simply  consist  of  rolling  the  crown  gear  on  axis  OA  and 
the  pinion  on  axis  OB  together,  when,  if  the  latter  were  formed  of 
a  plastic  material,  the  teeth  of  the  crown  gear  would  produce 
in  its  smaller  mate  corresponding  tooth  spaces  and  teeth  of  the 
proper  shape. 

By  Shaping  or  Planing.  —  There  is  but  one  form  of  tooth  to 
which  the  planing  operation  of  molding-generating  is  adapted. 


FORMING  THE  TEETH  251 

This  is  the  form  in  which  the  crown  gear  has  teeth  with  plane 
sides,  which  may  be  cut  with  a  straight-sided  tool.  If  the  draw- 
ing of  an  involute  rack  were  wrapped  around  the  periphery  of  the 
disk  in  Fig.  3,  about  axis  AO,  and  the  tooth  outlines  thus  deter- 
mined used  in  teeth  vanishing  at  O,  in  the  plane  of  the  pitch  line, 
the  resulting  crown  gear  would  be  of  this  type.  In  other  words, 
it  is  Tredgold's  approximation  of  the  involute  system.  In  Fig.  8 
such  a  crown  gear  is  shown  combined  with  a  simple  mechanism 
for  making  use  of  the  planing  or  shaping  operation  in  the  molding- 
generating  process.  The  gear  being  cut  is  keyed  on  a  loosely 
revolving  spindle,  to  which  is  also  keyed  a  master  gear,  formed 
on  the  same  pitch  cone  and  having,  in  this  case,  the  same  num- 
ber of  teeth.  This  spindle  is  so  set  in  relation  to  the  axis  about 
which  the  crown  gear  revolves,  that  the  master  gear  and  the 
crown  gear  mesh  together  properly,  the  crown  gear  being  of  the 
required  pitch,  and  having  the  proper  number  and  shape  of  teeth 
for  this  action.  If  now  the  crown  gear  be  rocked  about  its  axis, 
the  master  gear  will  also  rock  with  it,  carrying  the  gear  being  cut. 

The  blade  is  set,  as  shown  in  the  view  at  the  right,  so  that  its 
cutting  edge  coincides  with  the  plane  of  one  of  the  teeth  of  the 
crown  gear,  and  is  held  in  a  slide  which  guides  it  in  such  a  way 
that  it  moves  in  this  plane,  and  so  that  its  point  follows  the 
line  OX,  radiating  from  the  apex  0  of  the  pitch  cones.  The  tool 
will  evidently  represent  the  side  of  the  tooth  of  an  imaginary 
crown  gear,  which  is  adapted  to  mesh  properly  with  any  bevel 
gear  such  as  that  shown  being  cut,  keyed  to  the  master  gear  and 
having  the  same  pitch  cone  shape  and  number  of  teeth. 

If,  with  the  mechanism  so  arranged,  the  crown  gear  be  rotated 
so  as  to  start  the  cut  at  one  side  of  a  tooth  of  the  work  (which 
should  be  first  roughly  cut  to  size)  the  continued  rotation  of  the 
crown  gear  will  roll  the  master  gear  in  such  a  way  that  the  re- 
ciprocating blade  (representing  the  side  of  an  imaginary  crown 
tooth  meshing  with  the  work)  will  shape  the  side  of  the  tooth 
being  cut  to  the  proper  form,  by  the  molding-generating  process, 
on  the  same  principle  as  shown  in  Fig.  7. 

This  arrangement,  of  course,  is  not  a  practical  working  ma- 
chine as  shown,  since  there  is  no  provision  for  making  it  universal 


252  BEVEL  GEARING 

for  cutting  bevel  gears  of  other  pitch  cone  angles  and  numbers  of 
teeth,  or  for  indexing  the  work  with  relation  to  the  master  gear 
to  cut  the  remaining  teeth  of  the  work  shown  in  place.  Arranged 
as  shown,  however,  the  machine  will  cut  any  gear  within  its 
range  of  the  same  pitch  cone  angle  and  number  of  teeth  as  the 
master  gear.  To  cut  a  different  number  of  teeth  it  would  only 
be  necessary  to  alter  angle  XOY,  as  required,  setting  the  slide 
at  a  greater  angle  for  fewer  and  larger  teeth,  or  at  a  less  angle 
for  more  and  smaller  teeth. 

This  principle  will  be  found  applied  in  this  and  in  modified 
forms  in  various  machines  on  the  market.  One  of  the  modi- 
fications which  will  be  seen  is  equivalent  to  making  the  crown 
gear  in  Fig.  8  stationary,  and  swinging  the  frame  around  it  about 
axis  OA,  thus  rolling  the  master  gear  and  the  work  in  the  same 
relation  to  the  tool  as  when  the  frame  is  stationary  and  the  crown 
gear  is  revolved  as  just  described.  Still  another  possible  modi- 
fication would  consist  in  holding  the  master  gear  and  work  still, 
while  the  frame  is  swung  about  axis  OB.  In  this  case  the  crown 
gear  would  roll  on  the  master  gear,  rocking  the  tool  slide  in  such 
a  way  as  to  give  the  required  movement.  It  is  not  possible  to 
form  a  tooth  space  complete  with  a  single  tool,  as  shown  for  spur 
gears,  at  7\  in  Fig.  8,  Chapter  VIII,  without  cutting  the  tooth 
space  too  deep  at  the  outside  end.  A  separate  blade  has  to  be 
used  for  each  side  of  the  space  or  of  the  tooth. 

By  Milling,  and  by  Grinding  or  Abrasion.  —  Milling  cutters 
or  grinding  wheels  may  be  used  to  represent  the  space  of  the 
tooth,  as  they  represent  the  rack  tooth  for  spur  gears  in  Figs.  9 
and  10,  Chapter  VIII.  In  Fig.  9  of  the  present  chapter  is  shown 
diagrammatically  an  arrangement  by  which  two  cutters  or  grind- 
ing wheels  may  be  made  to  represent  the  two  sides  of  a  tooth  in 
such  a  way  that  by  them  a  tooth  space  may  be  finished  complete 
in  the  gear  to  be  cut  in  a  mechanism  similar  to  that  in  Fig.  8,  but 
without  requiring  the  reciprocating  movement.  The  same  diffi- 
culty arises  as  in  spur  gears,  of  the  center  of  the  tooth  being 
cut  in  deeper  than  the  ends,  owing  to  the  circular  form  of  the 
cutter.  This,  however,  makes  no  change  in  the  action  of  the 
finished  gear. 


FORMING  THE  TEETH 


253 


The  variety  of  applications  for  these  various  principles  and 
methods  of  operation  is  fully  as  great  in  bevel  gears  as  in  spur 
gears,  and  the  machines  in  which  they  are  incorporated  apply 
these  principles  and  methods  in  an  even  more  ingenious  fashion. 

Machines  for  Cutting  the  Teeth  of  Bevel  Gears  by  the  Formed 
Tool  Process.  —  A  very  common  method  of  using  the  formed 
tool  for  cutting  bevel  gears  makes  use  of  the  ordinary  plain  or 
universal  milling  machine  and  adjustable  dividing  head.  The 
process  of  cutting  bevel  gears  by  this  method  is  described  in 
detail  in  the  next  chapter. 


(IMAGINARY  CROWN  GEAR,  FORMING 
I  THE  BLANK  BY   THE  MOLDING- 

QENERATINQ     PROCESS. 


DISKS  REPRESENTING  ACTING 
FACES  OF  GRINDING  WHEELS  OR 

MILLING  CUTTERS;  THEY  COIN- 
CIDE WITH  THE  PLANE  FACES  OF 
A  TOOTH  OF  THE  CROWN  GEAR. 


Machinery 


Fig.  9.  Diagram  suggesting  Arrangement  of  Milling  Cutters  or  Grinding 
Wheels  for  Forming  the  Teeth  of  Bevel  Gears  by  the  Molding-generating 
Process 

Most  builders  of  automatic  gear  cutting  machines  furnish 
them,  if  desired,  in  a  style  which  permits  the  swiveling  of  the 
cutter  slide  or  of  the  work  spindle  to  any  angle  from  o  to  90  de- 
grees, thus  permitting  the  automatic  cutting  of  bevel  gears  by  the 
formed  cutter  process.  In  some  machines  of  this  type  the  cutter 
slide  is  mounted  on  an  adjustable  swinging  sector.  As  explained 
in  the  next  chapter,  it  is  necessary  when  cutting  bevel  gears,  to 
cut  first  one  side  of  the  teeth  all  around  and  then  the  other. 
Between  the  two  cuts  the  relation  of  the  work  and  cutter  to  each 
other,  as  measured  in  a  direction  parallel  to  the  axis  of  the  cutter 


254  BEVEL  GEARING 

spindle,  has  to  be  altered.  In  the  automatic  machine  this  is 
effected  by  shifting  the  cutter  spindle  axially  when  the  second 
cut  around  on  the  other  side  of  the  teeth  is  taken.  Suitable 
graduations  are  provided  for  the  angular  and  longitudinal 
adjustments. 

Bevel  Gear  Templet  Planing  Machines.  —  The  templet  planing 
machine  most  commonly  used  in  this  country  is  the  Gleason 
machine.  The  tool  is  carried  by  a  holder  reciprocated  by  an 
adjustable,  quick-return  crank  motion.  The  slide  which  carries 
this  tool-holder  may  be  swung  in  a  vertical  plane  about  the 
horizontal  axis  on  which  it  is  pivoted  to  the  head,  which  carries 
the  whole  mechanism  of  tool-holder,  slide,  crank,  driving  gearing, 
etc.  This  head,  in  turn,  may  be  swung  on  a  vertical  axis  about 
a  pivot  in  the  bed.  Circular  ways  guide  this  movement.  The 
intersection  of  the  vertical  and  horizontal  axes  of  adjustment 
(which  takes  place  in  mid-air  in  front  of  the  tool-slide)  is  the 
point  0  in  Fig.  5  where  the  templet  principle  is  shown  in  diagram- 
matic form.  The  blank  is  mounted  on  a  spindle  carried  by  a 
head  which  is  adjustable  in  and  on  the  top  of  the  bed  of  the  ma- 
chine so  that  the  apex  of  the  cone  of  the  gear  may  be  brought  to 
point  0  by  means  of  the  gages  which  are  a  part  of  the  equipment 
of  the  machine. 

Three  templets  are  used,  mounted  in  a  holder  attached  to  the 
front  of  the  bed  on  the  Gleason  bevel-gear  planing  machine.  The 
first  of  these  templets  is  for  " stocking"  or  roughing  out  the  tooth 
spaces.  It  guides  the  tool  to  cut  a  straight  gash  in  each  tooth 
space,  removing  most  of  the  stock.  After  each  tooth  space  has 
been  gashed  in  this  fashion,  the  templet  holder  is  revolved  to 
bring  one  of  the  formed  templets  into  position,  and  a  tool  is  set 
in  the  holder  so  that  its  point  bears  the  same  relation  to  the  shape 
of  the  tooth  desired  as  the  cam  roll  does  to  the  templet.  The 
head  is  again  fed  in  by  swinging  it  around  its  vertical  axis,  during 
which  movement  the  roll  runs  up  on  the  stationary  templet, 
swinging  the  tool  about  its  horizontal  axis  in  such  a  way  as  to 
duplicate  the  desired  form  on  the  tooth  of  the  gear.  One  side  of 
each  tooth  being  thus  shaped  entirely  around,  the  holder  is  again 
revolved  to  bring  the  third  templet  into  position.  This  has  a 


FORMING  THE  TEETH 


255 


reverse  form  from  the  preceding  one  adapted  to  cutting  the  other 
side  of  the  tooth.  A  tool  with  a  cutting  point  facing  the  other 
way  being  inserted  in  the  holder,  each  tooth  of  the  gear  has  its 
second  side  formed  automatically,  as  before,  completing  the  gear. 
The  swinging  movement  for  feeding  the  tool  and  the  indexing  of 
the  work  are  taken  care  of  by  the  mechanism  of  the  machine 
without  attention  on  the  part  of  the  operator. 

Adjustable  Former  for  Bevel  Gear  Planing.  —  A  convenient 
method  of  cutting  ordinary  bevel  gears  by  the  use  of  a  compara- 


Machinery 


Fig.  10.    Planing  Gears  of  Different  Angles  with  the  same  Former 

tively  small  number  of  formers  is  described  in  the  following 
paragraphs.  Bearing  in  mind  the  fact  that  to  a  given  circle  there 
corresponds  one  and  only  one  shape  of  involute,  one  can  readily 
see  by  referring  to  Fig.  10  that  a  pair  of  formers,  one  for  the  upper 
and  one  for  the  lower  side  of  the  tooth,  would  serve  for  all  gears 
if  they  could  be  set  at  any  desired  distance,  #,  from  the  apex  of 
the  pitch  cone.  If  the  shape  of  the  former  is  the  same  as  that  of 
a  gear  tooth  whose  pitch  radius  is  R,  it  will  be  suitable  for  cutting 
the  bevel  gear  indicated  by  a  full  section,  as  the  curvature  of  the 
gear  tooth  will  be  reduced  from  the  curvature  of  the  former  in  the 


256  BEVEL   GEARING 

same  proportion  as  R  is  to  r\  but  a  bevel  gear  of  any  other  pitch 
cone  angle  and  number  of  teeth,  for  instance  the  one  shown  in 
part  only,  having  a  pitch  cone  angle  AI,  can  be  cut  with  the  same 
former,  if  only  this  former  be  set  in  the  new  pitch  cone  at  such  a 
distance,  H\,  from  the  apex,  that  the  new  pitch  radius,  R,  be  the 
same  as  it  was  before.  The  number  of  teeth  in  either  of  the 
gears  is  immaterial  so  long  as  the  templet  is  long  enough.  A  long 
tooth  will  use  the  whole  of  the  templet,  while  a  shorter  tooth  will 
only  use  a  part  of  the  former. 

As  stated,  it  would  be  possible  for  one  former  to  cover  the 
whole  range  of  pitch  cone  angles  A,  but  since  on  any  given  ma- 
chine, distance  H  has  but  a  limited  variation,  this  necessitates 
a  series  of  formers  in  order  to  include  all  the  gears  capable  of 
being  cut  on  the  machine.  Suppose  we  have  a  machine  on  which 
a  former  can  be  set  between  30  and  45  inches  from  the  apex. 
Let  H  and  HI  in  Fig.  10  represent  these  two  extremes  of  distance, 

jy 

respectively.     It  is  apparent  from  this  diagram  that  —  =  tan  A. 

H 

If  2  inches  is  the  smallest  value  for  R  to  be  used  on  this  machine 
we  can,  by  using  it  in  the  above  formula  with  different  values  of 
H  between  30  and  45,  obtain  the  corresponding  values  of  A  which, 
when  laid  out  on  the  diagram,  Fig.  u,  will  be  represented  by  the 
curve  cd.  This  diagram  has,  however,  been  extended,  giving  a 
minimum  value  to  H  of  20  inches  and  a  maximum  value  of  55 
inches.  In  a  similar  way  all  the  other  curves  are  found,  the 
values  of  R  for  each  succeeding  one  being  chosen  so  that  each 
curve  intersects  the  45-inch  line  at  about  the  same  value  for  the 
pitch  angle  that  the  preceding  curve  intersects  the  30-inch  line, 
thus  always  covering  the  field  between  30  inches  and  45  inches, 
the  assumed  limits  of  the  machine. 

Take,  for  example,  a  bevel  gear  with  a  pitch  angle  of  30  de- 
grees; according  to  the  diagram  the  2i-inch  former,  or  a  former 
made  for  a  radius  R  =  21  inches,  is  the  one  to  be  used,  and  the 
reading  of  the  diagram  shows  that  it  should  be  set  about  365 
inches  from  the  apex.  If  the  machine  allows  a  shorter  or  longer 
adjustment  of  the  former  than  that  assumed  above,  the  3i|-inch 
former  at  about  54  inches  or  the  14-inch  former  at  24!  inches 


FORMING  THE  TEETH 


257 


from  the  apex  would  give  the  same  tooth  form.  When  the  pitch 
radius  of  the  former  exceeds  200  inches  the  involute  for  any  ordi- 
nary pitch  of  tooth  is  practically  a  straight  line,  and  a  former 
laid  out  accordingly  may  be  set  at  any  distance  from  the  apex. 
In  the  above  remarks  involute  formers  only  have  been  con- 
sidered. Owing  to  the  fact  that  the  cycloidal  curves  vary  not 


R=230"  155' 


105' 


70' 


47" 


31.5 


14"  9.5"  4.5"2' 


90°         80° 


70C 


50°  40° 

PITCH  ANGLE 


Xachinei'y 


Fig.  ix.    Diagram  for  Selecting  Formers 

only  with  the  pitch  radius,  but  with  the  pitch  as  well,  and  con- 
sequently  with  the  number  of  teeth  in  the  gear,  a  simple  diagram 
as  shown  above  cannot  be  obtained  for  cycloidal  formers. 

The  scheme  described  allows  the  use  of  a  smaller  number  of 
formers  than  would  otherwise  be  necessary  and  practically  makes 
allowance  for  the  errors  that  would  be  introduced  in  cutting 
a  gear,  in  which  the  pitch  angle  is  about  half-way  between 
those  of  the  two  nearest  formers  in  the  usual  way.  So  far  as 


258  BEVEL   GEARING 

the  author  knows,  planers  to  which  this  idea  may  be  applied  are 
not  being  built  at  the  present  time  in  such  a  way  that  the  dis- 
tance from  the  former  to  the  apex  is  adjustable;  but  there  are 
many  gear  planers  in  use  to  which  the  idea  can  be  applied  al- 
though they  are  of  an  old  design.  In  the  later  machines,  dimen- 
sion H  in  Fig.  10  is  constant  for  any  given  machine,  and  the 
formers  are  made  to  fit  this  dimension,  being  cut  in  a  generating 
machine  by  a  milling  cutter,  on  a  spindle  which  is  pivoted  to 
swing  about  the  apex  of  the  pitch  cone  in  the  same  way  that  the 
tool  slide  does. 

Bevel  Gear  Generating  Machines.  —  The  mechanism  illus- 
trated in  outline  in  Fig.  8  is  one  that  has  been  employed  in  a 
number  of  interesting  and  ingenious  machines.  The  first  appli- 
cation of  this  principle  was  made  by  Mr.  Hugo  Bilgram  of  Phila- 
delphia, Pa.  This  form  of  machine  in  the  hand-operated  style 
has  been  used  for  many  years.  A  more  recently  developed 
automatic  machine  of  the  same  type  has  also  been  built.  In  this 
the  movements  operate  on  the  same  principle  as  in  Fig.  8,  al- 
though in  a  modified  form.  Instead  of  rotating  the  crown  gear 
and  master  gear  together,  the  imaginary  crown  gear  and,  conse- 
quently, the  tool,  remain  stationary  so  far  as  angular  position 
is  concerned,  while  the  frame  is  rotated  about  the  axis  of  the 
crown  gear,  thus  rolling  the  master  gear  on  the  latter  and  rolling 
the  work  in  proper  relation  to  the  tool.  Instead  of  using  crown 
and  master  gears,  however,  a  section  of  the  pitch  cone  of  the 
master  gear  is  used,  which  rolls  on  a  plane  surface,  representing 
the  pitch  surface  of  the  crown  gear.  The  two  surfaces  are  pre- 
vented from  slipping  on  each  other  by  a  pair  of  steel  tapes, 
stretched  so  as  to  make  the  movement  positive.  A  still  further 
change  consists  in  extending  the  work  arbor  down  beyond  center 
O  in  Fig.  8,  mounting  the  blank  on  the  lower  side  of  the  center  so 
that  the  tool,  being  also  on  the  lower  side,  is  turned  the  reverse 
from  that  shown  in  the  diagram.  As  explained,  a  tool  with  a 
straight  edge  is  used,  representing  the  side  of  a  rack  tooth,  and 
this  tool  is  reciprocated  by  a  slotted  crank,  adjustable  to  vary  the 
length  of  the  stroke,  and  driven  by  a  Whitworth  quick-return 
movement.  The  feed  of  the  machine  is  effected  by  swinging  the 


FORMING  THE  TEETH  259 

frame  in  which  the  work  spindle  and  its  supports  are  hung  about 
the  vertical  axis  of  the  imaginary  crown  gear. 

As  stated,  the  machine  is  automatic.  The  operator  sets  the 
machine  and  places  a  previously  gashed  blank  on  the  work  spindle 
and  starts  the  tool  in  operation.  The  mechanism  provided  will, 
without  further  attention,  complete  one  side  of  all  the  teeth. 
The  machine  may  then  be  readjusted  and  the  tool  set  for  cutting 
the  other  side,  which  will  be  finished  in  the  same  automatic 
fashion.  The  mechanism  does  not  operate  on  the  principle  of 
completing  one  side  of  one  tooth  before  going  to  the  next.  It 
follows  the  plan  of  indexing  the  work  for  each  stroke  of  the  tool, 
the  rolling  action  being  progressive  with  the  indexing  so  as  to 
finish  all  the  teeth  at  once. 

The  Gleason  generating  machine  differs  from  the  previous 
machine  in  employing  two  tools,  one  on  each  side  of  the  tooth. 
The  construction  is  identical  with  the  mechanism  shown  in  Fig.  8, 
in  having  the  axes  of  the  tool-slides  and  of  the  blank  fixed  in  rela- 
tion to  each  other  during  the  operation,  the  tool-holders  and  the 
blank  rocking  about  their  axes  to  give  the  rolling  movement  for 
cutting.  The  rocking  is  effected  by  means  of  segments  of  an 
actual  crown  gear  and  master  gear.  The  segment  of  the  crown 
gear  is  permanently  attached  to  the  face  of  the  rear  of  the  cutter 
slide  frame,  while  the  segment  of  the  master  gear  (of  which  there 
are  several  furnished  with  the  machine,  the  one  used  being  chosen 
to  agree  with  the  angle  of  the  gear  to  be  cut)  is  clamped  to  the 
semi-circular  arm  pivoted  at  the  outer  end  of  the  machine  at  one 
side,  and  fastened  to  the  work  spindle  sleeve  on  the  other.  This 
arm  is  rocked  by  a  cam  mechanism  and  slotted  link. 

The  cycle  of  operations  is  as  follows:  The  machine  being  ad- 
justed properly  in  its  preliminary  position,  the  tool-slide  and  the 
head  on  which  it  is  mounted  are  swung  back  about  the  vertical 
axis  so  that  the  tools  clear  the  work.  The  blank  being  set  in  the 
proper  position,  a  cam  movement  swings  the  cutter  slide  head 
inward  until  the  reciprocating  tools  reach  the  proper  depth.  The 
cam  movement  first  mentioned  now  rocks  upward  the  semi- 
circular arm  extending  around  the  front  of  the  machine,  rolling 
the  blank  and  (through  the  segmental  crown  and  master  gears) 


260  BEVEL  GEARING 

the  slide,  until  the  tools  have  been  rolled  out  of  contact  in  one 
direction,  partially  forming  the  teeth  as  they  do  so.  The  arm  is 
then  rolled  back  to  the  central  position  and  along  downward  to 
the  lower  position,  until  the  tools  are  rolled  out  of  contact  with 
the  tooth  in  this  direction,  completing  the  forming  of  the  proper 
shape  as  they  do  so.  The  cam  then  rocks  the  arm  back  to  the 
central  position,  where  the  cutter-slide  head  is  swung  back  to 
clear  the  tooth,  and  the  work  is  indexed,  after  which  this  cycle  of 
operations  is  continued  for  the  next  tooth.  It  will  be  seen  that 
by  starting  from  the  central  position,  going  to  each  extreme 
and  returning,  all  parts  of  each  tooth  are  passed  over  twice,  giv- 
ing a  roughing  and  a  finishing  chip.  The  machine  is  entirely 
automatic. 


CHAPTER  XIII 
CUTTING   THE   TEETH   OF   BEVEL   GEARS 

SPECIAL  directions  for  operating  are  furnished  by  the  makers  of 
molding-generating  and  templet  planing  machines.  As  these 
directions  are  usually  adequate,  and  apply  only  to  the  particular 
machines  for  which  they  are  given,  this  chapter  will  be  confined 
to  giving  instructions  for  cutting  teeth  by  the  formed  tool  method 
only,  as  performed  on  the  milling  machine. 

Cutting  Bevel  Gears  in  the  Milling  Machine.  —  The  first 
requirement  for  setting  up  the  milling  machine  to  cut  bevel  gears 
is  a  true-running  blank,  with  accurate  angles  and  diameters.  If 
such  a  blank  cannot  be  found  in  the  lot  of  gears  to  be  cut,  it  will 
be  necessary  to  turn  up  a  dummy  out  of  wood  or  other  easily 
worked  material.  Otherwise  the  workman  is  inviting  trouble, 
whatever  his  method  of  setting  up. 

Fig.  i  shows  in  diagram  form  the  relative  positions  of  the  cutter 
and  the  work.  The  spindle  of  the  dividing  head  is  set  at  the 
cutting  angle,  as  shown,  and  the  cutter  (which  has  been  centered 
with  the  axis  of  the  work-spindle)  is  sunk  into  the  work  to  the 
whole  depth  W,  as  given  by  the  working  drawing. 

The  Brown  &  Sharpe  Mfg.  Co.  recommends  that  for  shaping 
with  a  formed  cutter,  the  cutting  angle  be  determined  by  sub- 
tracting the  addendum  angle  from  the  pitch  cone  angle,  instead  of 
subtracting  the  dedendum  angle.  In  other  words,  the  clearance 
at  the  bottom  of  the  tooth  is  made  uniform,  as  shown  in  Fig.  3, 
instead  of  tapering  toward  the  vertex.  This  gives  a  somewhat 
closer  approximation  to  the  desired  shape. 

Setting  the  Cutter.  —  The  centering  may  be  done  by  mount- 
ing a  true  hardened  center  in  the  taper  hole  of  the  spindle,  and 
lining  up  its  point  with  the  mark  which  will  be  found  inscribed 
either  on  the  top  or  on  the  back  face  of  the  tooth  of  the  com- 
mercial gear  cutter.  Setting  the  cutter  to  the  whole  depth  W  of 

261 


262  BEVEL  GEARING 

tooth  to  be  cut  is  effected  by  passing  the  work  back  and  forth 
under  the  revolving  cutter  and  slowly  raising  it  until  the  teeth 
of  the  cutter  just  bite  a  piece  of  tissue  paper  laid  over  the  edge 
of  the  blank.  This  must  be  done  after  centering.  The  dial  on  the 
elevating  screw  shaft  is  set  at  zero  in  this  position,  and  then  the 
knee  is  raised  an  amount  equal  to  the  whole  depth  of  the  tooth, 
reading  the  dial  from  zero.  This  is  evidently  not  exactly  right, 
since  the  measurement  should  be  taken  in  the  direction  of  the 
back  edge  of  the  tooth,  which  inclines  from  the  perpendicular  an 
amount  equal  to  the  dedendum  angle,  as  shown  in  Fig.  i.  In 
practice,  the  slight  difference  in  the  value  for  the  whole  depth 
thus  obtained  is  negligible. 

Having  thus  mounted  the  work  at  the  proper  angle  and  having 
thus  centered  the  cutter  and  set  it  to  depth,  two  tooth  spaces 
should  next  be  cut,  with  the  indexing  set  by  the  tables  furnished 
with  the  dividing  head  to  give  the  number  of  teeth  required  for 
the  gear.  Cutting  these  two  spaces  leaves  a  tooth  between  on 
which  trial  cuts  are  to  be  made  until  the  desired  setting  is  ob- 
tained. The  relative  positions  of  the  cutter  and  the  work  and 
the  shape  of  the  cuts  thus  produced  are  shown  in  the  upper  part  of 
Fig.  i.  It  will  be  seen  at  once  that  this  does  not  cut  the  proper 
shape  of  tooth.  As  explained  in  Chapter  XII,  all  the  elements 
of  the  bevel  gear  tooth  vanish  at  O,  the  vertex  of  the  pitch  cone  — 
that  is  to  say,  the  outer  corners  of  the  tooth  space  should  con- 
verge at  0  instead  of  at  A ,  and  the  sides  of  the  tooth  spaces  at  the 
bottom,  instead  of  having  the  parallel  width  given  them  by  the 
formed  cutter,  should  likewise  vanish  at  0.  Our  next  problem 
is  that  of  so  re-setting  the  machine  that  we  can  cut  gear  teeth  as 
nearly  as  possible  like  the  true  tooth-form  in  which  the  elements 
converge  at  0. 

Offsetting  and  Rolling  the  Blank  to  Approximate  the  Shape 
of  Tooth.  —  There  are  a  number  of  ways  of  approximating  the 
desired  shape  of  bevel  gear  teeth.  Of  these  we  have  selected  as 
most  practicable  the  one  in  which  the  sides  of  the  tooth  at  the 
pitch  line  converge  properly  toward  the  vertex  of  the  pitch  cone. 
Gears  cut  by  this  process  will  show,  of  course,  the  proper  thick- 
ness at  the  pitch  line  when  measured  by  the  gear  tooth  caliper  at 


CUTTING  THE  TEETH 


263 


either  the  large  or  the  small  ends.  This  method  of  approxima- 
tion produces  tooth  spaces  which,  at  the  small  end,  are  somewhat 
too  wide  at  the  bottom  and  too  narrow  at  the  top,  or,  in  other 
words,  the  teeth  themselves  at  the  small  end  are  too  narrow  at 


Machinery 


Fig.  i.    Relative  Positions  of  the  Formed  Cutter  to  the  Blank  when 
taking  a  Central  Cut 

the  bottom  and  too  wide  at  the  top.  To  make  good  running 
gears  they  must  be  filed  afterward  by  hand,  as  described  later. 
When  so  filed  they  are  better  than  milled  gears  cut  by  other 
methods  of  approximation  which  omit  the  hand  filing. 


264 


BEVEL   GEARING 


Set-over.  —  In  the  upper  part  of  Fig.  2  is  shown  a  section  of 
the  gear  in  Fig.  i,  taken  along  the  pitch  cone  at  PO.  It  will  be 
seen  that  the  teeth  at  the  pitch  line  converge,  but  meet  at  a  point 
considerably  beyond  the  vertex  0.  What  we  have  to  do  is  to 
move  the  cutter  off  the  center,  so  that  it  will  cut  a  groove,  one  side 
of  which  would  pass  through  0  if  extended  that  far.  The  amount 
by  which  the  cutter  is  set  off  the  center  is  known  as  the  "set- 


ONE  CENTRAL  CUT  TAKEN 


f  AO  [CENTER  LINES  OF  CENTRAL  CUTS 

IBO> 


1ST  OFFSET  OUT 
2ND       «.  n 


TWO  OFFSET  CUTS  TAKEN 


Machinery 


Fig.  2.    Section  on  the  Pitch  Cone  Surface  PO  of  Fig.  i  showing 
Central  and  Offset  Cuts 

over."  We  may  take,  for  instance,  for  trial  a  set-over  equal  to 
5  or  6  per  cent  of  the  thickness  of  the  tooth  at  the  large  end. 
Move  the  face  of  the  trial  tooth  away  from  the  cutter  by  the 
amount  of  this  trial  set-over,  having  first,  of  course,  run  the  cutter 
back  out  of  the  tooth  space.  Now  rotate  the  dividing  head 
spindle  to  bring  this  tooth  face  back  to  the  cutter  again,  stopping 
it  where  the  cutter  will  about  match  with  the  inner  end  of  the 
space  previously  cut.  Take  a  cut  through  in  this  position. 


CUTTING  THE  TEETH  265 

Trial  Cuts.  —  Having  proceeded  thus  far,  index  the  work  to 
bring  the  cutter  into  the  second  tooth  space  and  move,  the  blank 
over  to  a  position  the  other  side  of  the  central  position  by  an 
amount  equal  to  the  same  set-over,  thus  moving  the  opposite 
face  of  the  trial  tooth  away  from  the  cutter.  Rotate  the  dividing 
head  spindle  again  to  bring  this  face  toward  the  cutter  until  the 
latter  matches  the  central  space  already  cut  at  the  inner  end  of 
the  teeth.  Take  the  cut  through  in  this  position. 

Now  with  vernier  gear  tooth  calipers  or  with  fixed  gages  ma- 
chined to  the  proper  dimensions  measure  the  thickness  of  the 
tooth  at  the  pitch  line  at  both  large  and  small  ends.  (See  Fig.  15, 
Chapter  I.)  If  the  thickness  is  too  great  at  both  the  large  and 
the  small  ends,  rotate  the  tooth  against  the  cutter  and  take 
another  cut  until  the  proper  thickness  at  either  the  large  or  small 
end  has  been  obtained.  If  the  thickness  comes  right  at  both 
ends  the  amount  of  set-over  is  correct.  If  it  is  right  at  the  large 
end  and  too  thick  at  the  small  end  the  set-over  is  too  much.  If 
it  is  right  at  the  small  end  and  too  thick  at  the  large  end  the  set- 
over  is  not  enough.  The  recommended  trial  set-over  (5  or  6  per 
cent  of  thickness  of  the  tooth  at  the  pitch  line  at  the  large  end) 
will  probably  not  be  enough,  so  two  or  three  cuts  will  have  to  be 
taken  on  each  side  of  the  trial  tooth,  as  described,  before  the 
proper  amount  is  found. 

Having  found  the  proper  set-over,  the  cross-feed  screw  is  set  to 
that  amount  and  the  cut  is  taken  clear  around  the  gear.  Then 
the  cross-feed  screw  is  set  to  give  the  same  amount  of  set-over 
the  other  side  of  the  center  line  and  the  work  is  rotated  until  the 
cutter  matches  the  tooth  spaces  already  cut  at  the  small  end  and 
is  run  through  the  work.  The  tooth  will  generally  be  found  too 
thick,  so  the  work  spindle  is  rotated  still  more  until  the  tooth  is 
of  the  proper  thickness,  when  the  gear  is  again  cut  clear  around 
on  this  second  cut. 

The  number  of  holes  it  was  necessary  to  move  the  index  pin  on 
the  dividing  plate  circle  between  the  first  and  the  second  cuts  to 
get  the  proper  thickness  of  tooth,  should  be  recorded.  On  suc- 
ceeding gears  it  will  thus  only  be  necessary  to  take  a  first  cut  clear 
around  with  the  work  set  over  by  the  required 'amount  on  one  side 


266 


BEVEL  GEARING 


of  the  center  line,  and  then  a  second  cut  around  with  the  work 
set  over  on  the  other  side  of  the  center  line,  rotating  the  index 
crank  the  number  of  holes  necessary  to  give  the  proper  thickness 
of  tooth  between  the  cuts. 

It  will  be  noted  that  the  shifting  of  the  blank  by  the  index 
crank  is  only  used  for  bringing  the  thickness  of  tooth  to  the 
proper  dimension.  In  some  cases,  particularly  in  gears  of  fine 
pitch  and  large  diameter,  this  adjustment  will  not  be  fine 
enough  —  that  is  to  say,  one  hole  in  the  index  circle  will  give 
too  thick  a  tooth  and  the  next  one  too  thin  a  tooth.  To  sub- 
divide the  space  between  the  holes,  most  dividing  heads  have  a 
fine  adjustment  for  rotating  the  worm  independently  of  the 
crank.  Every  milling  machine  should  be  provided  with  such  an 
adjustment. 

In  large  gears  it  is  best  to  take  the  central  cuts  shown  in  Fig.  i 
clear  around  every  blank  before  proceeding  with  the  approximate 
cuts.  This  gives  the  effect  of  roughing  and  finishing  cuts,  and 
produces  more  accurate  gears.  The  central  cuts  may  be  made 
in  a  separate  operation  with  a  roughing  or  stocking  cutter  if  de- 
sired. It  might  also  be  mentioned  that  it  is  common  practice  to 
turn  up  a  wooden  blank  for  making  the  trial  cuts  shown  in  Figs, 
i  and  2,  to  avoid  the  danger  of  spoiling  the  work  by  mistakes  in 
the  cut-and-try  grocess. 

Positive  Determination  of  the  Set-over.  —  This  cut-and-try 
process,  however,  may  be  practically  eliminated  by  calculating 
the  set-over  from  the  following  table  and  formula: 

Table  for  Obtaining  Set-over  for  Cutting  Bevel  Gears 


Ratio  of  Pitch  Cone  Radius  to  Width  of  Face  (-  \ 

No.  of 

Cutter 

3 

3H 

3H 

3% 

4 

4M 

4W 

494 

5 

5^ 

6 

7 

8 

1 

i 

0.254 

0.254 

0.255 

0.256 

0.257 

0.257 

0.257 

0.258 

0.258 

0.259 

0.260 

0.262 

0.264 

2 

0.266 

0.268 

0.271 

0.272 

0.273 

0.274 

0.274 

0.275 

0.277 

0.279 

o  280 

0.283 

0.284 

3 

0.266 

0.268 

0.271 

0.273 

0.275 

0.278 

0.280 

0.282 

0.283 

0.286 

0.287 

0.290 

0.292 

4 

0.275 

0.280 

0.285 

0.287 

0.291 

0.293 

0.296 

0.298 

0.298 

0.302 

0.305 

0.308 

0.311 

5 

0.280 

0.285 

0.290 

.0.293 

0.295 

0.296 

0.298 

0.300 

0.302 

0.307 

0.309 

0.313 

0.315 

6 

0.311 

0.318 

0.323 

0.328 

0.330 

0.334 

0.337 

0.340 

0.343 

0.348 

0.352 

0.356 

0.362 

7 

0.289 

0.298 

0.308 

0.316 

0.324 

0.329 

0.334 

0-338 

0.343 

0.350 

0.360 

0.370 

0.376 

8 

0.275 

0.286 

0.296 

0.309 

0.319 

0.331 

0.338 

0.344 

0.352 

0.361 

0.368 

0.380 

0.386 

CUTTING  THE  TEETH  267 

For  obtaining  the  set-over  by  the  above  table,  use  this  formula : 

Tc      factor  from  table  ,  ^ 

Set-over  =  -£  -  -        -j-  .     .     .     (i) 

P  =  diametral  pitch  of  gear  to  be  cut; 

Tc  =  thickness  of  cutter  used,  measured  at  pitch  line. 

• 

Given  as  a  rule  this  would  read :  Find  the  factor  in  the  table  corre- 
sponding to  the  number  of  the  cutter  used  and  to  the  ratio  of  pitch 
cone  radius  to  width  of  face;  divide  this  factor  by  the  diametral  pitch, 
and  subtract  the  result  from  half  of  the  thickness  of  the  cutter  at  the 
pitch  line. 

As  an  illustration  of  the  use  of  this  table  in  obtaining  the  set- 
over,  we  will  take  the  following  example:  A  bevel  gear  of  24 
teeth,  6  pitch,  30  degrees  pitch  cone  angle  and  i \  face  is  to  be  cut. 
These  dimensions,  by  the  rules  given  in  Chapter  X,  call  for  a  No.  4 
cutter  and  a  pitch  cone  radius  of  4  inches. 

In  order  to  get  the  factor  from  the  table,  we  must  know  the 

ratio  of  pitch  cone  radius  to  width  of  face.    This  ratio  is  -i-  =  — 

1.25       i 
gi 
or  about  2i .     The  factor  in  the  table  for  this  ratio  with  a  No.  4 

cutter  is  0.280.  We  next  measure  the  cutter  at  the  proper  depth 
of  5  +  A  for  6  pitch,  which  is  found  in  the  column  marked  "depth 
of  space  below  pitch  line"  in  a  regular  table  of  tooth  parts,  or 
by  dividing  1.157  by  the  diametral  pitch.  This  gives  5  +  A  = 
0.1928  inch.  We  find,  for  instance,  that  the  thickness  of  the 
cutter  at  this  depth  is  o.  1 745  inch.  The  dimension  will  vary  with 
different  cutters,  and  will  vary  in  the  same  cutter  as  it  is  ground 
away,  since  formed  bevel  gear  cutters  are  commonly  provided 
with  side  relief.  Substituting  these  values  in  the  formula,  we 
have: 

c  ,  O-I745      0.280  .,  .    , 

Set-over  =  — Llr  —  _ —  =  0.0406  inch 

26  , 

which  is  the  required  dimension. 

The  work  must  now  be  set  off  center  on  one  side  of  the  cutter 
by  this  amount,  taking  the  usual  precautions  to  avoid  errors  from 
back-lash.  In  this  position  the  cutter  is  run  through  the  blank, 
the  latter  being  indexed  for  each  tooth  space  until  it  has  been  cut 


268  BEVEL  GEARING 

around.  If  a  central  or  roughing  cut  has  been  previously  taken 
as  suggested  in  an  earlier  paragraph,  it  will  be  necessary  to  line 
up  this  cut  at  the  small  end  of  the  tooth  with  the  cutter.  This  is 
done  by  rotating  the  tooth  space  back  toward  the  cutter,  either 
by  moving  the  index  crank  as  many  holes  in  the  dial-plate  as  are 
necessary,  or  by  means  of  such  other  special  provisions  as  may 
be  made  for  doing  this  in  the  index  head,  independently  of  the 
dial-plate. 

Having  thus  cut  one  side  of  the  tooth  to  proper  dimensions, 
the  work  must  be  set-over  by  the  same  amount  the  other  side  of 
the  position  central  with  the  cutter,  taking  the  same  precautions 
in  relation  to  back-lash  as  before,  and  rotating  the  blank  to  again 
line  up  the  cutter  with  the  tooth  space  at  the  small  end  of  the 
tooth.  With  this  setting,  take  a  trial  cut  as  already  explained. 
This  will  be  found  to  leave  the  tooth  whose  side  is  trimmed  in 
this  operation  a  little  too  thick,  if  the  cutter  is  thin  enough,  as  it 
ought  to  be,  to  pass  through  the  small  end  of  the  tooth  space  of 
the  completed  gear.  This  trial  tooth  should  now  be  brought  to 
the  proper  thickness  by  rotating  the  blank  toward  the  cutter, 
moving  the  crank  around  the  dial  for  the  rough  adjustment,  and 
bringing  it  to  accurate  thickness  by  such  means  as  may  be 
provided  in  the  head.  In  the  Brown  &  Sharpe  head,  this  fine 
adjustment  is  effected  by  two  thumb-screws  near  the  hub  of  the 
index  crank,  which  turn  the  index  worm  with  relation  to  the 
crank. 

Testing  for  Correctness  of  the  Setting.  —  With  reference  to 
the  use  of  the  table  and  formula,  the  Brown  &  Sharpe  Mfg.  Co., 
after  trial  in  their  gear  cutting  department,  say:  "We  feel  fairly 
confident  it  is  within  working  limits  of  being  satisfactory." 
While  this  sounds  encouraging,  it  will  evidently  be  wise  to  be 
sure  we  are  right  before  going  ahead,  as  the  slight  approxima- 
tions involved  in  the  derivation  of  the  formula  (to  be  explained 
later)  may  bring  the  setting  not  quite  right,  so  that  the  thickness 
of  the  tooth  at  the  large  and  the  small  ends  is  not  what  it  ought 
to  be.  This  point  may  be  tested  by  measuring  the  tooth  at  both 
the  large  and  the  small  ends  with  a  vernier  caliper,  the  caliper 
being  set  so  that  the  addendum  at  the  small  end  is  in  the  proper 


CUTTING  THE  TEETH  269 

proportion  to  the  addendum  at  the  large  end  —  that  is  to  say, 

C  —  F 

that  it  is  in  the  ratio  — — — .     In  taking  these  measurements, 

C 

if  the  thicknesses  at  both  the  large  and  the  small  ends,  which 
should  be  in  this  same  ratio,  are  too  great,  rotate  the  tooth  toward 
the  cutter  and  take  another  cut  until  the  proper  thickness  at 
either  the  large  or  small  end  has  been  obtained.  As  already 
mentioned  in  connection  with  the  ordinary  cut-and-try  method, 
if  the  thickness  is  right  at  the  large  end  and  too  thick  at  the  small 
end,  the  set-over  is  too  much.  If  it  is  right  at  the  small  end  and 
too  thick  at  the  large  end,  the  set-over  is  not  enough,  and  should 
be  changed  accordingly,  as  is  done  by  the  regular  "cut-and-try" 
process.  The  formula  and  table  herewith  given,  however,  ought 
to  bring  it  near  enough  right  the  first  time,  and  in  the  general  run 
of  work  it  can  be  safely  relied  on. 

Use  of  the  Formula  for  Other  Methods  of  Correction.  —  It  is 
customary  also  among  workmen  expert  in  cutting  bevel  gears 
with  formed  cutters,  to  disregard  rules  and  formulas  for  the  selec- 
tion of  the  cutters,  and  depend  on  their  experience  to  get  shapes 
which  require  somewhat  less  filing  than  would  otherwise  be 
necessary.  Whenever  this  dependence  on  experienced  an  judg- 
ment requires,  as  it  sometimes  does,  the  use  of  a  cutter  of  finer 
pitch  than  that  of  the  teeth  of  the  bevel  gear  at  the  large  end, 
the  values  given  in  the  table  are  inapplicable.  The  following 
formula  may  then  be  used: 

Set-over  =  -c  -  ^-^  X  £ 

2  2  F 

in  which  te  is  the  thickness  of  the  cutter  measured  at  a  depth 
s  +  A,  obtained  as  shown  in  Fig.  i.  This  has  been  tried  on 
several  widely  varying  cases  with  good  results.  It  requires,  it 
will  be  seen,  two  measurements  of  the  cutter  in  place  of  the  single 
one  required  when  the  regular  pitch  of  cutter  is  used. 

Filing  the  Teeth.  —  The  method  of  cutting  bevel  gears  just 
described  requires  the  filing  of  the  points  of  the  teeth  at  the  small 
end.  This  can  be  done  "by  the  eye"  very  skillfully  when  the 
workman  is  used  to  it.  The  operation  consists  in  filing  off  a 
triangular  area  extending  from  the  point  of  the  tooth  at  the  large 


270  BEVEL  GEARING 

end  to  the  point  at  the  small  end,  thence  down  to  the  pitch  line 
at  the  small  end  and  back  diagonally  to  the  point  at  the  large 
end  again.  This  is  shown  in  Fig.  4,  by  the  shaded  outline. 
Enough  is  taken  off  at  the  small  end  of  the  tooth  so  that  the 
edges  of  the  teeth  at  the  top  appear  to  converge  at  vertex  0  in 
Figs,  i  and  2. 

Testing  Bevel  Gear  Teeth.  —  The  bevel  gears  may  be  tested 
for  the  accuracy  of  the  cutting  and  filing  by  mounting  them  in 
place  in  the  machine  and  revolving  them  at  high  speed,  or  by 
mounting  them  in  a  testing  machine  made  for  the  purpose.  The 
marks  of  wear  produced  by  running  them  together  under  pres- 
sure, with  the  back  faces  flush  with  each  other,  should  extend  the 


achinery 


Fig.  3.    Parallel  Clearance  best  adapted     Fig.  4.    Shaded  Area  showing 
to  Shaping  with  Formed  Cutter  Part  Corrected  by  Filing 

whole  length  of  the  tooth  at  the  pitch  line.  If  it  does  not,  the 
amount  of  set-over  allowed  in  cutting  them  was  at  fault,  being 
too  little  if  they  bear  heavily  at  the  large  ends,  and  too  much  if 
they  bear  heavily  at  the  small  ends.  The  bearing  area  should 
also  be  fairly  evenly  distributed  over  the  sides  of  the  teeth  above 
the  pitch  line,  from  the  large  to  the  small  end.  If  it  is  not,  the 
filing  is  at  fault.  The  marks  of  wear  will  not  in  any  case  extend 
far  below  the  pitch  line  in  a  pinion  of  few  teeth. 

It  is  possible  to  get  along  without  filing  by  decreasing  the 
amount  of  set-over  so  as  to  make  the  teeth  too  thin  at  the  pitch 
line  at  the  small  end,  when  they  are  of  the  right  thickness  at  the 
large  end.  This  does  not  give  as  good  running  gears,  however, 
as  when  the  method  just  described  is  followed. 


CUTTING  THE  TEETH  271 

Cutting  Bevel  Gears  on  the  Automatic  Gear-cutting  Machine. 

—  The  directions  for  cutting  bevel  gears  on  the  milling  machine 
apply  in  modified  form  to  the  automatic  gear-cutting  machine  as 
well.  The  set-over  is  determined  in  the  same  way,  but  instead 
of  moving  the  work  off  center,  the  cutter  spindle  is  adjusted 
axially  by  means  provided  for  that  purpose.  Some  machines  are 
provided  with  dials  for  reading  this  movement.  The  cutter  is 
first  centered  as  in  the  milling  machine,  and  then  shifted  —  first 
to  the  right,  and  then  to  the  left  of  this  central  position. 

The  rotating  of  the  work  to  obtain  the  proper  thickness  of 
tooth  is  effected  by  unclamping  the  indexing  worm  from  its 
shaft  (means  usually  being  provided  for  this  purpose)  and 
rotating  the  worm  until  the  gear  is  brought  to  proper  posi- 
tion. Otherwise  the  operations  are  the  same  as  for  the  milling 
machine. 

Derivation  of  Formula  for  Positive  Set-over.  —  The  derivation 
of  the  formula  and  the  method  of  calculating  the  table  need  not 
concern  the  man  who  merely  desires  to  use  them,  as  he  can 
employ  them  with  no  knowledge  of  mathematics  other  than  that 
required  for  plain  subtraction  and  division.  For  those,  however, 
who  desire  to  understand  the  origin  of  the  table,  the  following 
explanation  will  be  interesting. 

Fig.  2  shows  a  section  such  as  would  be  made  by  turning  off 
the  bevel  gear  blank  down  to  the  pitch  cone  —  in  other  words,  it 
is  a  section  on  the  conical  surface  PO  of  Fig.  i.  The  same  refer- 
ences apply  to  both  figures.  We  find,  if  we  take  a  cut  with  the 
cutter  set  central,  that  the  side  of  the  tooth  space  will  not  pass 
through  the  vertex  0,  but  through  some  point  0'  at  one  side. 
The  distance  between  0'  and  O  is  X,  which  is  the  amount  by 
which  the  cutter  will  have  to  be  offset  to  bring  the  side  of  the 
tooth  space  at  the  pitch  line  radial  with  vertex  0.  A  formula 
can  be  derived  by  simple  proportion  to  obtain  this  offset,  in  terms 
of  Tc,  tc,  F  and  C.  The  formula  is: 

'Tc  ~  /. . 

4 

This  determination  of  the  set-over,  of  course,  involves  one  or 


272  BEVEL  GEARING 

two  approximations  of  minor  importance,  which  will  be  readily 
perceived  from  an  examination  of  the  diagrams. 

While  this  formula  seems  to  furnish  a  means  for  obtaining  by 
measurement  and  calculation  the  amount  of  set-over,  it  is  rather 
clumsy.  It  remains  therefore  to  put  it  in  more  usable  form. 

T 

From  an  examination  of  the  formula,  we  note  that  while  —  is 

2 

a  variable,  depending  on  the  thickness  of  the  cutter,  the  quantity 
in  parenthesis  remains  constant  as  the  cutter  grows  thinner  from 
being  ground  down.  In  fact,  by  taking  the  measurements  Tc 
and  tc  on  a  one-diametral  pitch  cutter,  and  calling  them  Tc'  and 
//  it  would  be  possible  to  put  this  in  the  form: 

Tc      i  /T!-tcr 

-^~P(~^ 

T  '  —  / '      C 

in  which  the  quantity  — X  —  would  be  constant  for  all 

2  r 

£ 

cases  of  all  pitches  having  the  same  ratio  of  —  and  using  the 

F 

same  number  of  cutter.       •  ^ 
Now  it  is  possible  to  put  this  formula  in  still  simpler  form  by 

T '  —  t f      C 

tabulating  the  values  of   — X  — ,  as  measured  on  a  one 

2  r 

C 

pitch  cutter,  for  different  values  of  -  .    This  has  been  done  in 

r 

c 

the  table  for  thirteen  values  of  —  ,  which  cover  the  major  part  of 

r 

the  bevel  gears  cut  by  the  formed  tool  process.  Using  the  factor 
as  given  in  the  table,  the  formula  reads: 

„                        Tc     factor  from  table  /  N 

X  =  set-over  =  — .     .     .     (i) 

as  we  have  already  given  it. 

The  method  of  filling  in  the  table  will  be  easily  understood. 
A  one-pitch  cutter  is  measured  with  the  Brown  &  Sharpe  gear- 
tooth  caliper  at  depth  S  +  A  for  dimension  Te',  and  at  depth 
s  +  A  for  //  (see  Fig.  i).  Of  course,  s  +  A  and  consequently  // 

C  T>    i— 

will  vary,  as  —  is  taken  - ,  —  ,  etc.,  respectively.    Having  found 


CUTTING  THE  TEETH  273 

these  dimensions,  the  quantities  to  use  in  the  table  are  evidently 
obtained  by  the  formula: 

Factor  from  table  =  TC  ~  tc  X  i     ....     (4) 

2  r 

The  form  of  table  and  formula  here  given  is  also  suitable  for 
recording  for  future  use  the  amount  of  set-over  obtained  by  the 
"  cut-and-try "  process  for  other  methods  of  approximation  than 
that  here  given.  Transposing  Formula  (i)  to  solve  for  the  factor, 
we  have: 

Factor  from  table  =  P\~ °  ~  set-over  J    .     .     .     (5) 

in  which  the  measurement  as  before  is  taken  on  the  cutter  used 
for  the  work  in  hand.  By  recording  the  factors  for  all  cases 
met  with,  and  thus  gradually  filling  in  the  tables  from  his  own 
practice,  the  machinist  would  be  able  to  put  the  data  in  usable 
form  to  apply  to  future  jobs. 

Practicability  of  the  Milling  Process.  —  The  methods  obtained 
with  the  templet  process  and,  above  all,  with  the  generating 
process,  are  so  much  superior  to  those  obtained  with  the  milling 
cutter  that  the  use  of  the  latter  should  be  avoided  wherever 
possible.  It  has  a  legitimate  field,  however,  on  gears  too  small 
to  be  cut  on  any  commercial  planing  machine.  In  general,  it  is 
not  considered  advisable  to  plane  gears  having  teeth  finer  than 
10  to  12  diametral  pitch.  It  is  allowable,  however,  to  mill  gears 
of  coarser  pitch  which  are  to  run  at  slow  speeds,  or  which  are  to 
be  used  only  occasionally  —  such,  for  instance,  as  the  bevel  gears 
used  for  turning  the  elevating  screws  of  a  planer  cross-rail,  or 
those  used  in  connection  with  other  hand-operated  mechanisms. 
Under  ordinary  conditions  it  is  impracticable  to  mill  bevel  gears 
having  teeth  coarser  than  3  diametral  pitch,  no  matter  what  the 
service  for  which  they  are  to  be  used. 

Approximations  Involved  in  Positive  Set-over  Calculations.  — 
Two  of  the  approximations  involved  in  this  method  may  be 
mentioned.  In  Fig.  i,  it  will  be  noticed  that  the  various  dimen- 
sions W,  S  +  -4j  etc.,  are  taken  perpendicular  to  the  bottom  of 
the  tooth,  instead  of  perpendicular  to  the  pitch  line,  as  they 
should  be.  The  setting  of  the  cutter  to  depth  by  this  usual 


274 


BEVEL   GEARING 


method,  therefore,  involves  an  error,  but  it  is  so  slight  as  to  be 
negligible.  The  other  approximation  relates  to  the  use  of  the 
uncorrected  tooth  thickness  and  addendum  for  measuring  the 
cutters  in  preparing  the  table.  The  chordal  tooth  thickness 
might  have  been  used,  but  at  the  cost  of  a  considerable  compli- 
cation in  the  process.  It  was  found  by  investigation  that  this 
refinement  would  not  affect  the  final  result  enough  to  make  it 
worth  while. 


Machinery 


Fig.  5.    Method  of  setting  Bevel  Gear  Cutter 

Another  Positive  Method  of  Setting  Bevel  Gear  Cutter.  — 

Another  method  of  setting  the  cutter  for  milling  quiet-running 
bevel  gears  is  outlined  in  the  following.  In  using  this  method, 
two  cuts  are  required.  The  set-up  for  taking  the  first  cut  is 
shown  in  Figs.  5  and  6,  while  Figs.  5  and  7  illustrate  the  set-up 
for  the  second  cut.  As  Fig.  5  shows,  the  cutting  angle  equals  the 
pitch  cone  angle  of  the  gear.  The  gear-cutter  is  set  by  a  pointer  P 
which  is  adjusted  so  that  the  end  coincides  with  the  apex  of  the 
pitch  cone.  After  this  pointer  is  set,  the  gear-cutter  is  adjusted 
until  a  line  on  the  side  of  the  tooth,  representing  the  pitch  circle, 


CUTTING  THE  TEETH 


275 


coincides  with  the  end  of  the  pointer.  The  required  number 
of  tooth  spaces  is  then  milled,  after  which  the  lateral  position 
of  the  cutter  is  changed  as  shown  in  Fig.  7 ;  that  is,  the  pointer 
is  set  to  coincide  with  the  pitch  circle  on  the  opposite  side  of  the 
cutter.  The  teeth  are  then  finished  by  taking  a  second  series 
of  cuts,  as  explained  later. 

In  order  to  locate  the  pitch  circle  on  the  cutter,  a  little  blue 
vitriol  is  placed  on  one  of  the  cutter  teeth  and  a  pair  of  dividers 
is  used  to  mark  the  arc  of  the  pitch  circle  AB  on  this  tooth,  after 


Machinery 


Fig.  6.    Method  of  setting  Gear  Cutter  for  taking  First  Cut 

a  centering  plug  has  been  inserted  in  the  bore  of  the  cutter. 
The  radius  R  of  the  pitch  circle  is  obtained  by  subtracting  2t 
from  the  outside  diameter  of  the  cutter  and  dividing  the  result 
by  2.  (The  distance  /  or  depth  of  space  below  the  pitch  line 
equals  1.157  divided  by  the  diametral  pitch.) 

After  marking  the  pitch  circle  on  the  cutter,  the  latter  is 
mounted  on  the  arbor  of  the  milling  machine.  The  gear  blank 
G  is  next  mounted  on  mandrel  M  of  the  dividing  head  and  is 
inclined  to  the  pitch  cone  angle.  There  is  a  hole  f  inch  in  diam- 
eter and  about  i  inch  deep  in  the  center  of  the  mandrel  M  and 
pointer  P  fits  into  this  hole.  The  pointer  can  be  moved  in  and 


276 


BEVEL   GEARING 


out  by  hand,  but  is  tight  enough  to  remain  in  any  position.  By 
placing  a  straightedge  against  the  face  of  the  gear  blank,  in  two 
or  three  different  positions,  and  sliding  pointer  P  in  or  out,  as 
may  be  found  necessary,  its  sharp  end  can  be  made  to  coincide 
with  the  axis  of  the  cone. 

The  milling  machine  table  is  next  adjusted  vertically,  to  the 
right  or  left,  and  laterally  until  the  end  of  the  pointer  coincides 
with  the  pitch  line  on  one  side  of  the  milling  cutter,  as  indicated 
in  Figs.  5  and  6.  When  this  has  been  done,  the  table  is  moved 


Machinery 


Fig.  7.    Method  of  setting  Gear  Cutter  for  taking  Second  Cut 

to  the  left  and  the  pointer  removed  from  the  mandrel,  after  which 
the  first  cut  is  taken.  When  the  cutter  has  been  located  by  this 
method,  the  pitch  line  of  the  tooth  face  KL  coincides  with  an 
element  of  the  cone.  After  the  first  tooth  has  been  cut,  the  suc- 
ceeding teeth  are  cut  by  indexing  the  gear  blank  in  the  usual 
manner. 

The  pin  is  now  reinserted  in  the  mandrel  and  its  sharp  end 
again  located  at  the  apex  of  the  cone.  The  table  is  then  moved 
until  the  relative  position  of  the  pin  and  cutter  are  as  shown  in 
Figs.  5  and  7,  the  cutter  being  set  on  the  opposite  side  of  the 
center  line.  In  Fig.  7,  the  dotted  trapezoid  shows  the  position 


CUTTING  THE  TEETH  277 

occupied  by  the  cutter  during  the  first  series  of  cuts,  and  the  full 
line  its  position  for  the  second  or  finishing  cuts. 

The  table  is  now  moved  to  the  left,  the  pin  removed  and  the 
dividing  head  turned  to  the  left  through  an  angle  corresponding 
to  one  tooth  space  or  180  degrees  divided  by  the  number  of  teeth. 
The  machine  is  now  ready  for  taking  the  second  cut,  which  is 
usually  very  light,  seldom  exceeding  0.012  to  0.015  inch.  With 
the  work  located  in  this  way,  the  pitch  line  of  tooth  face  ON  also 
coincides  with  the  surface  of  the  pitch  cone.  Fig.  5  shows  the 
amount  X  by  which  the  cut  taken  by  this  method  exceeds  the 
correct  depth  at  the  inner  ends  of  the  teeth. 


CHAPTER  XIV 
LONG   AND    SHORT   ADDENDUM    GEARS 

Object  of  Gears  with  Lengthened  Addendum.  —  The  bevel 
driving  gears  in  the  rear  axle  have  probably  given  more  trouble 
to  automobile  makers  and  users  than  any  other  two  gears  used  on 


STANDARD  14^  INVOLUTE 
14  TOOTH  PINION 


LONG  ADDENDUM  14j|  INVOLUTE 
14  TOOT'H  PINION 


Machinery,  N.  Y. 


Fig.  i.    Comparison  between  Standard  and  Long  Addendum  Gears 

the  car.  For  that  reason  any  system  of  gear  tooth  design  that 
tends  to  quieter  running,  greater  strength  or  durability  is  de- 
serving of  consideration.  The  system  described  in  the  following 
is  not  new,  although  none  of  the  authors  of  standard  gear  books 
in  the  past  have  deemed  it  worthy  of  more  than  a  passing  com- 

278 


LONG  ADDENDUM   GEARS 


279 


ment.  It  is,  of  course,  understood  that  the  tooth  is  of  true 
involute  or  of  octoid  form,  depending  upon  whether  it  is  produced 
for  a  spur  or  bevel  gear.  The  special  feature  of  it  is  the  lengthen- 
ing of  the  addendum  of  the  pinion  tooth,  with  a  corresponding 
shortening  of  the  addendum  of  the  gear  tooth,  the  whole  depth 
remaining  the  same  as  in  the  standard  tooth.  The  tables  and 

Table  I.    Chordal  Thickness  of  Teeth  For  Spur  Gears,  i  Diametral  Pitch, 
Special  Pitch  Depth 

To  obtain  chordal  thickness  of  teeth  and  corrected  pitch  depth  for  any 
diametral  pitch  other  than  i,  divide  figures  in  table  by  diametral  pitch 
required. 

2o-degree  Pressure  Angle 


For  Pinions,  Addendum  =  6.7  Working  Depth 

Number  of  Teeth 

Chordal  Thickness 

Corrected  Pitch  Depth 

12 

.8545 

.4720 

13 

•8554 

.4665 

14 

•8567 

.4618 

I5-I6 

•8573 

•4559 

I7-I8 

•8584 

•4495 

IQ-2O 

.8592 

•  4442 

21-22 

.8597 

.4402 

23-25 

.8601 

•  436i 

26-29 

.8606 

•  4316 

30-34 

.8609 

.4272 

For  Gears,  Addendum  =  0.3  Working  Depth 

35-  41 

.2792 

0.6107 

42-  54 

•2793 

0.6085 

55-  79 

•2794 

o  .  6060 

80-134 

•2795 

o  .  6040 

134 

•2795 

o  .  6030 

For  bevel  gears,  find  chordal  thickness  of  tooth  and  corrected  pitch  depth  of  gear  with  the 
same  number  of  teeth  as  a  spur  gear  having  a  diameter  equal  to  twice  the  back  cone  distance. 

formulas  given  in  the  following  are  abstracted  from  a  paper 
read  by  Mr.  E.  W.  Weaver  before  the  Society  of  Automobile 
Engineers. 

With  the  addendum  or  face  of  the  driver  lengthened,  the  arc 
of  approach  of  the  gear  tooth  action  is  lessened  and  the  arc  of 
recess  is  increased  —  becoming  all  recess  and  no  approach  when 
the  driver  has  only  faces  and  the  driven  gear  only  flanks.  This 


280 


BEVEL     GEARING 


produces  particularly  smooth-running  gears  —  almost  equal,  in 
fact,  to  spiral  gears.  As  is  well  known,  the  friction  of  the  arc 
of  approach  is  much  greater  than  that  of  the  arc  of  recess  — 
something  on  the  principle  of  a  man  dragging  a  stick  after  him 
or  pushing  it  ahead  of  him. 

Pinions  with  Small  Number  of  Teeth. — Another  advantage 
of  this  system  is  the  very  great  improvement  in  the  shape  of  the 
tooth  when  the  pinion  has  a  small  number  of  teeth.  It  is  readily 
seen  in  Fig.  i  that  the  pinion  teeth  with  the  long  addendum  are 
fully  as  strong  as  the  gear  teeth,  while  with  the  standard  tooth 
they  are  not.  This  being  the  case,  it  is  possible,  in  designing  a 

Table  H.    Diametral,  Circular  and  Metric  Pitches 


Diametral 
Pitch 

Circular 
Pitch 

Nearest 
Metric  Pitch 
or  "Module" 

Diametral 
Equivalent  of 
"  Module  " 

Circular  Pitch 
Corresponding 
to  Module 

2H 

1.3962 

II  .O 

2.309 

1.3607 

2M 

1.2566 

IO.O 

2.540 

1.2370 

2H 

I  .  1424 

9.0 

2.822 

I-H33 

3 

1.0472 

8.0 

3-175 

0.9896 

3H 

0.8976 

7.0 

3-628 

0.8659 

4 
4H 

0.7854 
0.6981 

6.0 

5-5 

4-233 
4.618 

0.7422 
o  .  6803 

5 

0.6283 

S-o 

5.080 

0.6185 

SH 

0-5712 

4-5 

S-644 

0.5566 

6 

0.5236 

4.0 

6.350 

0.4948 

rear  axle  drive,  to  select  a  smaller  number  of  teeth  for  the  pinion 
than  could  otherwise  be  used;  for  instance,  if  the  number  of 
teeth  previously  used  had  been  17  and  54,  the  combination  of  15 
and  48  would  give  almost  exactly  the  same  ratio  and  would  be 
fully  as  strong.  With  a  5^-inch  diameter  pitch  circle,  the  outside 
diameter  of  the  large  gear  would  be  decreased  somewhat  over 
\  inch,  thereby  making  the  case  that  much  smaller  and  lighter, 
and  gaining  advantages  all  around. 

The  disadvantage  of  using  a  small  number  of  teeth  in  a  pinion 
with  the  standard  tooth  has  been  the  small  amount  of  stock  left 
between  the  bore  of  the  shaft  and  the  bottom  of  the  tooth  spaces. 
This  is  eliminated  to  a  large  extent  by  the  decrease  of  the  deden- 
dum  of  the  pinion  tooth. 


LONG  ADDENDUM   GEARS 


28l 


CHORDAL 


Field  of  Application.  —  The  field  of  application  is  limited  to 
gear  sets  having  a  large  difference  in  the  number  of  teeth  in  the 
gears  and  pinions,  on  account  of  the  gear  tooth  becoming  weaker 
as  the  number  of  teeth  decreases.  The  gear  having  the  lengthened 
addendum  must  at  all  times  be  the  driver,  as  in  reversing  the 
application  of  power,  the  arc  of  recess  becomes  the  arc  of  ap- 
proach with  its  greater  friction. 
This  is  of  no  account  in  the 
driving  gears  of  a  car,  as  in 
coasting  no  power  is  transmitted, 
except  when  there  is  a  propeller- 
shaft  brake. 

Efficiency.  —  One  point  which 
has  not  been  touched  on  is  the 
efficiency,  and  consequent  life, 
of  a  gear  of  this  system  as  com- 
pared with  one  of  the  ordinary 
or  of  the  stub  form  of  tooth. 
No  exhaustive  scientific  tests  to 
determine  this  have  been  made, 
to  the  author's  knowledge,  so  it 
is  a  matter  for  further  demon- 
stration. 

Finding  Circular  Thickness 
of  Tooth.  —  The  addendum  of 
the  pinion  tooth,  of  the  type 
described,  is  arbitrarily  taken  as 
0.7  of  its  working  depth,  and 
0.3  for  the  gear  tooth,  for  both 
14^-  and  2o-degree  pressure  angles.  To  find  the  circular 
thickness  of  the  tooth  at  the  pitch  line  at  these  depths,  mul- 
tiply the  circular  pitch  by  0.5659  for  the  pinion,  and  by  0.4341 
for  the  gear  for  i4^-degree  pressure  angle.  For  2o-degree  pres- 
sure angle  multiply  the  circular  pitch  by  0.5927  and  0.4073, 
respectively,  for  the  pinion  and  gear. 

The  impossibility  of  getting  accurate  circular  measurements 
necessitates  the  calculation  of  the  chordal  thickness  and  cor- 


Maohinery,  N.Y. 


Fig.  2.    Nomenclature  of  Gear  Tooth 


282 


BEVEL  GEARING 


Table  HI.    Formulas  for  Long  Addendum  Bevel  Gears 


Name 

Symbol 

Formula 

Pinion 

Gear 

Pinion 

Gear 

Number  of  teeth 

Ni 

Pd 

PC 

Dl 

Pi 
W 
A, 
Ei 
G 
F 

h 
Oi 
Ti 

C 
Bi 
r\ 

Si 

ti 

Vi 

Hi 

Si 
Li 
& 

Nz 

Pd 

PC 

D2 

• 

p2 

W 
A, 
Ei 
G 
F 

h 
0, 
T2 

C 
B2 
r2 

S2 

t* 

V2 

H2 

S2 
L2 
K2 

Ni  =  PdXDi 

Pd-Nl-  N* 
Fd  ~  Di  ~  D* 

Table  II 

*H8 

**-8 

2 

Pd 
Ai  =  o.7X  W 

Ei  =  A2  +  G 
G  =  PCX  0.05 
F=  W+G 

I\  =  AI  X  cos  pi 
Oi  =  Di  +  2li 

Ti  =  Pc  X  0.5927 

c-      Dl 

N2=PdXDi 

PA  -  Nl  -  N* 

Fd  -  Di  ~  D* 
Table  II 

n      N* 
D*  =  Pd 

-*-t 

2 
Pd 

A2  =  o.3XW 
E2  =  Ai  +  G 
G  =  PCX  0.05 
F=  W+G 

/a  =  A2  X  cos  pi 
02  =  Di  +  2lt 
T2  =  Pc-Ti 

C  ^i— 

2  X  sin  pi 

B2=CXta.npt 
tanra  =  ^ 

tan  Si  =  -~ 

t-i.  =  p*  +  r2 
v-i  =  pz  —  s* 

H2  =  A  2  X  sin  p2 

S2  =  2  X  Pd  X  B2 
Table  I 
Table  I 

Diametral  pitch 

Circular  pitch  

Pitch  diameter  in  inches  . 
Pitch  angle     

Working  depth  

Addendum               

Dedendum 

Clearance     

Full  depth 

One-half  diameter  incre- 
ment 

Outside  diameter  

Circular  thickness  

Pitch  cone  distance  
Back  cone  distance 

2  X  sin  pi 
Bi  =  C  X  tan  pi 

tan  ri  =  -£ 

•pt 

tan  Si  =  -£ 
ti  =  pi  +  ri 

v  i  =  pi  -  si 

Hi  =  AiXsmpi 

Si  =  2XPdXBi 
Table  I 
Table  I 

Addendum  angle  

Dedendum  angle  
Face  angle 

Cutting  angle  

Distance  from  crown  to 
pitch  line  

No.  teeth  in  spur  gear 
having  diameter  equal 
to  twice  the  back  cone 
distance  

Chordal  thickness  
Corrected  pitch  depth.  .  . 

LONG  ADDENDUM   GEARS 


283 


rected  pitch  depth.     Referring  to  Fig.  2,  let  R  equal  the  pitch 
radius  for  a  spur  gear  or  the  back  cone  distance  for  a  bevel  gear. 

.      ,        =i       360  degrees 

"  2       (2  R  X  3.1416)  -*-  Circular  Thickness 

Chordal  thickness  =  2  X  sin  a  X  R. 

Corrected  pitch  depth  =  versed  sin  a  X  R  +  the  given  pitch 
depth. 


-JF 


Ah £— 


Machinery,  N.  Y. 


Fig.  3.    Symbols  used  in  Formulas  in  Table  III 

These  values  have  been  tabulated  for  gears  with  as  large  a 
range  of  tooth  numbers  as  the  system  is  applicable  to  with 
advantage.  (See  Table  I.)  The  formulas  required  have  also 
been  arranged  (in  Table  III)  in  the  logical  routine  order  for  all 
necessary  calculations  for  a  pair  of  bevel  gears  of  this  system; 
symbols  are  as  indicated  in  Fig.  3. 

As  some  firms  are  using  the  metric  pitch  or  "  Module "  sys- 
tem, Table  II  is  given  for  their  convenience. 


284  BEVEL  GEARING 

Duplication  of  Gears  having  Long  and  Short  Addenda.  —  The 

formula  according  to  which  the  gear  addendum  is  made  0.3  of 
the  working  depth,  and  the  pinion  addendum  0.7  of  the  working 
depth  of  the  tooth,  with  the  working  depth  equal  to  twice  the 
standard  addendum,  was  proposed  by  the  Gleason  Works  and 
may  be  considered  as  standard  for  this  class  of  gearing  in  this 
country.  In  the  examination  of  foreign-made  bevel-gear  drives 
using  this  style  of  tooth,  it  has  been  found,  as  explained  by  Mr. 
E.  W.  Baxter,  in  the  June,  1913,  number  of  MACHINERY,  that  the 
ratio  of  the  gear  addendum  to  the  pinion  addendum  varies  with 
each  set  of  gears  examined,  and  in  order  to  render  the  duplication 
of  such  gears  easier,  the  following  formulas  and  tables  were 
evolved.  In  order  to  explain  the  method,  we  will  assume  that 
a  set  of  bevel  gears  has  been  ordered  to  duplicate  samples  which 
are  measured  as  carefully  as  possible  and  the  dimensions  tabu- 
lated as  follows: 

Gear  Pinion 

Number  of  teeth  48  13 

Measured  outside  diameter  9.66    inches  3.14    inches 
Measured  face  angle                  76 J  degrees          i8j  degrees 
Measured  cutting  angle             71 J  degrees          13^  degrees 

Measured  depth  of  tooth  0.43    inch  0.43    inch 

The  number  of  teeth  on  the  pinion  being  odd,  the  outside 
diameter  is  found  by  adding  to  the  diameter  of  the  bore,  twice 
the  distance  from  the  edge  of  the  bore  to  the  tip  of  a  tooth. 

If  we  figure  the  outside  diameters  according  to  the  method 
used  for  standard  gears  and  find  that  the  outside  diameter  of 
the  pinion  is  larger  and  that  of  the  gear  smaller  than  the  dimen- 
sions of  the  gears  to  be  duplicated,  we  will  know  that  this  pair 
of  gears  has  been  designed  with  long  and  short  addendum  teeth. 
After  a  little  experience  it  will  not  be  necessary  to  figure  the 
standard  dimensions,  as  ordinarily  an  inspection  of  the  pinion 
tooth  is  all  that  is  necessary.  By  referring  to  Table  IV,  we  find 
that  the  nearest  standard  pitches  having  whole  depths  around 
0.43  inch  are  f-inch  circular  pitch  (whole  depth  0.429  inch)Jand 
5  diametral  pitch  (whole  depth  0.431  inch).  We  will  therefore 
assume  that  the  gears  are  5  diametral  pitch,  as  this  will  be  the 


LONG  ADDENDUM  GEARS  285 

easiest  dimension  to  use  in  the  calculations.  The  standard  out- 
side diameters  for  48  and  13  tooth,  5  pitch  gears  are  9.704  inches 
and  2.986  inches,  while  the  measured  diameters  are  9.66  inches 
and  3.14  inches.  As  the  measured  diameter  for  the  gear  is  smaller 
and  that  for  the  pinion  is  larger  than  the  standard  dimensions, 
we  may  safely  assume  that  the  gears  are  cut  with  long  and  short 
addenda.  As  the  center  angles  remain  the  same  as  in  the  stand- 
ard gears  (on  account  of  the  pitch  diameters  not  changing),  we 
have  only  to  look  them  up  in  a  handbook  on  gearing  or  calculate 
them  from  the  following  formula.  Calling  a  the  center  angle  of 

the  Sear:  Number  of  teeth  in  gear 

Tangent  a  =  — - 

Number  of  teeth  in  pinion 

Center  angle,  pinion  =  90  degrees  —  a. 

Calculating  for  48  and  13  teeth,  we  have: 

Tangent  a  =  £f  =  3.69230. 

Center  angle,  gear  =  74  degrees  51  minutes  =  a. 

Center   angle,   pinion  =  90    degrees  —  a  =  90   degrees  —  74 
degrees  51  minutes  =  15  degrees  9  minutes. 

Tabulating  the  known  quantities  and  representing  those  that 
are  unknown  by  suitable  letters,  we  have  the  following: 

Number  of  teeth  on  gear  48 

Number  of  teeth  on  pinion  13 

Center  angle  of  gear  74  deg.  51  min. 

Cosine  of  center  angle  of  gear  0.26135 

Center  angle  of  pinion  15  deg.  9  min. 

Cosine  of  center  angle  pinion  0.96524 

Measured  outside  diameter  of  gear  9.66  inches 

Measured  outside  diameter  of  pinion         3.14  inches 

Pitch  diameter  of  gear  D 

Pitch  diameter  of  pinion  d 

Addendum  of  gear  (short)  X 

Addendum  of  pinion  (long)  F 

Standard  addendum  of  gear  and  pinion  j  „ 
or  half  the  working  depth  j  * 

Z  also  equals  the  pitch  diameter  of  either  gear  divided  by  the 
number  of  teeth. 

Pressure  angle  14 \  degrees 


286  BEVEL   GEARING 

The  pressure  angle  may  be  found  by  taking  an  impression  of 
the  backs  of  the  large  gear  teeth  on  paper  and  measuring  with 
a  protractor  the  angle  which  the  side  of  the  tooth  makes  with  a 
line  drawn  through  the  center  of  the  tooth  in  the  direction  of  its 
length. 

The  formula  for  finding  the  outside  diameters  of  a  pair  of 
bevel  gears  when  the  addenda  are  unequal  in  gear  and  pinion  is 
as  follows:  Pitch  diameter  +  2  (special  addendum  X  cosine  of 
center  angle)  =  outside  diameter. 

Substituting  the  known  and  unknown  quantities  we  get : 

For  the  gear,  D  +  0.5227  X  =  9.66  inches  =  outside 

diameter (i) 

For  the  pinion,  d  +  1.93048  Y  =  3.14  inches  =  outside 

diameter (2) 

As  the  clearance  and  whole  depths  are  the  same  for  both  gear 
and  pinion  in  this  style  of  gearing,  the  long  addendum  and  the 
short  addendum  added  together  equal  the  working  depth  and 
also  equal  twice  the  "diameter  addendum"  2  Z. 

This  gives  us  a  third  equation. 

X  +  Y  =  2  Z    or    Z  =  *L±_T  (,) 

2 

As  previously  stated,  Z  is  the  standard  addendum  for  the 
depth,  and  Z  multiplied  by  the  number  of  teeth  in  each  gear  gives 
the  pitch  diameters  of  the  gears.  Therefore  we  derive  the 

following:  48Z  =  £> (4) 

^Z  =  d (5) 

Substituting  these  values  of  D  and  d  in  Equations  (i)  and  (2) 
we  get:  48  Z  +  0.5227  X  =  9.66 

Simplifying  Z  +  0.01089  X  =  0.20125  .....  (6) 
13  Z  +  1.930487  =  3.14 

Simplifying         Z  + 0.1485   7  =  0.24154      (7) 

Substituting  the  value  of  Z  from  Equation  (3)  in  Equations 
(6)  and  (7)  and  simplifying  we  get: 

h  0.01089  X  =  0.20125 

X  +  Y  +  0.02178  X  =  0.4025 


LONG  ADDENDUM   GEARS  287 

i.  02 1 78  X  +  7  =  0.4025 (8) 

-  +  0.14857  =  0.24154 

X  +  7  +  0.297  7  =  0.48308 

X  +  1.297  Y  =  0.48308 (9) 

Multiplying  Equation  (8)  by  1.297  and  subtracting  Equation 
(9)  we  get: 

1.32525  X  + 1.297  7  =  0.52204 
X  +  1.297  7  =  0.48308 

0.32525  X  =0.03896 

X  =  0.11978  inch  =  addendum 

for  gear  (short). 

Substituting  this  value  for  X  in  Equation  (9)  we  have: 

0.11978  +  1.297  ^  =  0.48308 
1.297  7  =  0.3633 

7  =  0.28010  inch  =  addendum  of  pinion 
(long). 

We  can  now  get  the  value  of  Z  by  substituting  both  X  and  7 
in  Equation  (3). 

0.11978  +  0.28010  =  2  Z 
0.39988  =  2  Z 

Z  =  0.19994  inch  =  standard  addendum. 

Z  is  probably  0.200  inch  and  the  pitch  for  this  depth  is  5  diam- 
etral pitch. 

X  =  o.i  20  inch  which  is  •£-$  of  0.400  inch  (whole  depth) 
7  =  0.280  inch  which  is  ^  of  0.400  inch 

as  per  the  Gleason  formula. 

Correction  for  Shrinkage  in  Hardening.  —  The  large  or  ring 
gear  is  apt  to  shrink  in  hardening,  the  amount  of  this  shrinkage 
depending  upon  the  size  of  the  gear,  the  kind  of  steel  and  the 
heat-treatment  used.  The  shrinkage  is  not  often  more  than  0.02 
inch,  but  even  this  amount  will  affect  the  teeth  of  the  gear  by 
reducing  the  circular  pitch,  and  if  this  shrinkage  is  not  compen- 
sated for  when  turning  and  cutting  the  gear,  the  pinion  will  not 
mesh  with  the  finished  ring  gear  at  the  same  place  that  it  did 


288 


BEVEL  GEARING 


Table  IV.    Addenda  for  Diametral  and  Circular  Pitches 

(Whole  depth  =  2.157  X  addendum) 


Addendum 
in  Inches 

Whole  Depth 
in  Inches 

1  Diametral 
Pitch 

Circular  Pitch 

Metric  Pitch 
Module 

Addendum 
in  Inches 

Whole  Depth 
in  Inches 

Diametral 
Pitch 

Circular  Pitch 

Metric  Pitch 
Module 

Fraction, 
Inches 

Decimal, 
Inches 

Fraction, 
Inches 

Decimal, 
Inches 

0.125 
0.126 
O.I27 
0.128 
0.129 
0.130 
0.131 
0.132 
0.133 
0.134 
0.135 
0.136 

0.137 
0.138 

0.139 
0.140 
0.141 
0.142 

0.143 
0.144 

0.145 
0.146 
0.147 
0.148 
0.149 
0.150 
0.151 
0.152 
0.153 
0.154 
0.155 
0.156 

0.157 
0.158 

0.159 
0.160 
0.161 
0.162 
0.163 
0.164 
0.165 
0.166 
0.167 
0.168 
0.169 
0.170 
0.171 
0.172 
0.173 

0.270 
0.272 
0.274 
0.276 
0.278 
0.280 
0.282 
0.285 
0.287 
0.289 
0.291 
0.293 
0.295 
0.298 
0.300 
0.302 
0.304 
0.306 
0.308 
0.3H 
0.313 
0.315 
0.317 
0.319 
0.321 
0.324 
0.326 
0.328 
0.330 
0.332 
0-334 
0-337 
0-339 
0.341 
0-343 
0-345 
0-347 
0-349 
0.352 
0-354 
0.356 
0.358 
0.360 
0.362 

0.365 
0.367 

0.369 
0.371 

0-373 

8 

... 

0.4000 

3-2 

0.174 

0.175 
0.176 
0.177 
0.178 
0.179 
O.lSo 
O.lSl 
0.182 
0.183 
0.184 
0.185 
0.186 
0.187 
0.188 
0.189 
0.190 
0.191 
0.192 
0.193 
0.194 

o.i95 
0.196 
0.197 
0.198 
0.199 

O.2OO 
O.2OI 
O.2O2 
0.203 
O.2O4 
0.205 
O.2O6 
O.2O7 
0.208 
0.209 
0.210 
0.2II 
O.2I2 
0.213 
O.2I4 
O.2I5 

0.216 
0.217 
0.218 
0.219 

O.22O 
O.22I 
O.222 

0.375 
0.378 
0.380 
0.382 

0.384 
0.386 
0.388 
0.390 
0.393 
0.395 
0.397 
0.399 

0.401 

0.403 

0.406 
0.408 
0.410 
0.412 
0.414 

0.416 

0.419 

0.421 

0.423 
0.425 
0.427 
0.429 

0.431 
0.434 
0.436 

0.438 

0.440 
0.442 
0.444 

0-447 
0.449 

o.45i 
0-453 
0-455 
0.457 
0-459 
0.462 
0.464 
0.466 
0.468 
0.470 
0.472 
0-475 
0-477 
0.479 

SK 

Mi 

% 
»A* 

0.5625 

4-5 

7M 

3-3 

5H 

"tf 

.... 

4.6 

7W 

M2 

3-4 

H 

Ma 

4-7 

7M 

3-5 

Me 

0.4375 

5M 

l^a 

4*8 

7 

63/4 

H 

Mi 
M« 

.... 

3^6 

3-7 

% 

O.6ooo 

•• 

.... 

4-9 

5 

% 
Yii 

0.6250 

5-0 
5-i 

.... 

3-8 

6H 

.  .  . 

.... 

3-9 

4H 

5-2 

6H 

H 

0.5000 

4-0 
4-i 

4K 

2^2 

5-3 

•- 

K 

0.6666 

5-4 

6 

1^2 

.... 

4.2 
4-3 

4H 

5-5 

•• 

iMa 

0.6875 

5*6 

•• 

... 

4-4 

4W 

.... 

LONG  ADDENDUM   GEARS 


289 


Table  IV.    Addenda  for  Diametral  and  Circular  Pitches.  —  (Continued) 

(Whole  depth  =  2.157  X  addendum) 


Addendum 
in  Inches 

}{ 

<u  G 

|.s 

Diametral 
Pitch 

Circular  Pitch 

ll 

o'd 
•CO 

tsE 

Addendum 
in  Inches 

Whole  Depth 
in  Inches 

Oo  •  •  •  •  I  Diametral 
$•••<•  1  Pitch 

Circular  Pitch 

Metric  Pitch 
Module 

II 

3£ 

£~ 

Decimal, 
Inches 

Fraction, 
Inches 

1« 

.§-5 
££ 

0.223 
0.224 
0.225 
O.226 
0.227 
0.228 
0.229 
0.230 
0.231 
0.232 
0.233 
0.234 
0-235 
0.236 
0.237 
0.238 
0.239 
0.240 
0.241 
0.242 
0.243 
0.244 
0.245 
0.246 

0.247 
0.248 
0.249 
0.250 
0.251 
0.252 
0.253 
0.254 
0.255 
0.256 
0.257 
0.258 
0.259 
0.260 
0.261 
0.262 
0.263 
0.264 
0.265 
0.266 
0.267 
0.268 
0.269 
0.270 
0.271 

0.481 
0.483 
0.485 
0.488 
0.490 
0.492 
0.494 

0.496 
0.498 

0.500 
0.503 
0.505 
0.507 
0.509 
0.511 

0.513 
0.516 
0.518 
0.520 
0.522 
0.524 
0.526 
0.529 

0-531 
0-533 
0-535 
0-537 
0-539 
0.541 
0.544 
0.546 
0.548 
0-550 
0-552 
0-554 
0-557 
0-559 
0.561 

0.563 
0-565 
0.567 
0.569 
0.572 

0.574 
0.576 
0.578 
0.580 
0.582 
0.585 

Vio 

o  .  7000 

5-7 

0.272 
0.273 
0.274 
0.275 
0.276 
0.277 
0.278 
0.279 
0.280 
0.281 
0.282 
0.283 
0.284 
0.285 
0.286 
0.287 
0.288 
0.289 
0.290 
0.291 
0.292 
0.293 
0.294 
0.295 
o.  296 

0.297 
0.298 
0.299 
0.300 
0.301 
0.302 
0.303 
0.304 

0.305 
0.306 
0.307 
0.308 
0.309 
0.310 
0.311 
0.312 

0.313 
0.314 
0.315 

0.316 
0.317 
0.318 
0.319 
0.320 

0.587 
0.589 

0-591 
0-593 
0-595 
0.598 
0.600 
0.602 
0.604 
0.606 
0.608 
o.6n 
0.613 
0.615 
0.617 
0.619 
0.621 
0.623 
0.626 
0.628 
0.630 
0.632 
0.634 
0.636 
0.639 
0.641 
0.643 
0.645 
0.647 
0.649 
0.651 
0.654 
0.656 
0.658 
0.660 
0.662 
0.664 
0.667 
0.669 
0.671 
0.673 
0.675 
0.677 
0.680 
0.682 
0.684 
0.686 
0.688 
0.690 

M 

6.9 

7-0 

4% 

M 

2%2 

'Ms 

5-8 

% 

0.8750 

7-i 

Mi 

5-9 

96 

7-2 

4M 

6.0 

3H 

Mo 

2%2 

'Mi 

0.9000 

7-3 

4H 

H 

0.7500 

6.1 

.... 

»Ma 

.... 

7-4 

.... 

6.2 

3% 

'Ms 
is/Is 

0-9375 

7-5 
7^6 

4 

ji 
'*fa 

.... 

6-3 

•• 

.... 

6-4 

7-7 

3J* 

ff 

**u 

Mi 

0.8000 
0.8125 

6^5 
6^6 

3H 

3^2 

.... 

7:s 

7.9 

56 

0*8333 

6^7 

•• 

... 

8.0 

3% 

2%2 



6^8 

3H 

I 

I.  OOOO 

8.1 

290 


BEVEL  GEARING 


before  the  ring  gear  was  hardened.  If  we  suppose  that  this 
shrinkage  was  not  taken  into  account  and  amounted  to  0.02  inch, 
the  measured  outside  diameter  of  the  large  gear  would  be  9.64 
inches  and  that  of  the  small  gear  3.14  inches,  no  appreciable 
shrinkage  being  evident  in  the  pinion  on  account  of  its  small  size. 
By  substituting  these  values  in  the  preceding  formulas,  we  can 
find  whether  or  not  the  correct  dimensions  can  be  found  under 
these  conditions.  If  this  calculation  is  made,  it  will  be  found 
that  the  small  amount  that  the  ring  gear  will  shrink  will  not  alter 
the  results  enough  to  cause  any  serious  error. 

Results  of  Calculation.  —  From  the  above  data,  we  can  figure 
the  correct  face  and  cutting  angles  and  outside  diameters,  using 
the  measured  values  to  check  the  calculations.  The  full  data  is 
tabulated  below: 

Long  and  Short  Addendum  Gears  with  i4§-Degree  Pressure  Angle 

Ratio,  3T9^  to  i 
Diametral  pitch,  5 
Working  depth,  0.4000  inch 
Whole  depth,  0.4314  jnch 


Addendum 

Dedendum 

Face  angle 

Center  angle 

Cutting  angle 

Pitch  diameter 

Outside  diameter 

Difference  between 
special  and  standard 
addendum  angles 

Difference  between 
Bilgram  system  and 
standard  addendum 
angles  for  the  above 
ratio 


Gear 

0.1200  inch 
0.3 1 14  inch 
76  deg.  14  min. 
74  deg.  51  min. 
71  deg.  1 6  min. 
9.600  inches 
9.662  inches 


55  minutes 


Pinion 

0.2800  inch 
0.1514  inch 
1 8  deg.  22  min. 
15  deg.    9  min. 
13  deg.  24  min. 
2.600  inches 
3.140  inches 


55  minutes 


59  minutes 


59  minutes 


CHAPTER  XV 

s 

SKEW  BEVEL  GEARS 

Characteristics  of  Skew  Bevel  Gears.  —  The  skew  bevel  gear 
is  a  special  form  sometimes  used  to  connect  a  pair  of  shafts 
which  are  not  parallel  and  which  do  not  intersect.  Skew  bevel 
gears  have  straight  teeth  which  bear  on  each  other  along  a 
straight  line;  a  plane  through  the  center  of  the  tooth,  however, 
intersects  the  axis  of  the  gear  instead  of  passing  through  the  axis 
as  in  ordinary  bevel  gearing.  The  difficulty  of  producing  cor- 
rectly shaped  teeth  in  skew  bevel  gears  has  been  the  most  impor- 
tant reason  why  these  gears  are  so  little  used,  and,  generally 
speaking,  the  spiral  gear  and  worm  gear  appear  to  be  the  most 
practical  solution  of  the  problem  of  connecting  by  a  single  pair 
of  gears  two  shafts  which  are  not  in  the  same  plane.  In  the 
following,  however,  one  method  for  producing  skew  bevel  gear 
teeth  will  be  described. 

Skew  Bevel  Gear  Tooth  which  can  be  Produced  by  the  Mold- 
ing-generating Process.  —  In  the  January,  1913,  number  of 
MACHINERY,  Mr.  J.  M.  Bartlett  proposes  a  method  for  generating 
the  teeth  of  skew  bevel  gears  by  the  molding-generating  process. 
This  method  is  based  on  the  assumption  that  Sang's  theory  holds 
true  for  skew  gears  as  well  as  for  spur  and  bevel  gears.  In  a 
system  of  interchangeable  involute  spur  gears,  each  gear  will 
mesh  perfectly  with  a  rack  the  pitch  surface  of  which  is  a  plane, 
and  the  teeth  of  which  can  each  be  swept  out  by  a  single  stroke 
of  a  straight-edged  planing  tool.  In  a  system  of  interchangeable 
bevel  gears  (octoidal  system),  each  gear  will  mesh  perfectly  with 
a  crown  gear  the  pitch  surface  of  which  is  a  plane  disk,  and  the 
teeth  of  which  can  each  be  swept  out  by  a  single  stroke  of  a 
straight-edged  planing  tool.  In  the  proposed  system  of  inter- 
changeable skew  gears,  each  gear  will  mesh  perfectly  with  a 
"rack"  the  pitch  surface  of  which  is  a  "right  helicoid,"  and  the 

291 


292  BEVEL  GEARING 

tooth  surfaces  of  which  are  hyperbolic  paraboloids  capable  of 
being  swept  out  by  a  single  stroke  of  a  straight-edged  planing 
tool.  The  teeth  in  these  gears  do  not  vanish  at  the  gorge,  do  not 
have  to  be  undercut  to  avoid  interference,  are  reversible,  and 
give  the  same  obliquity  of  action  throughout  their  length.  By 
using  sections  of  the  hyperboloids  some  distance  from  the  gorge, 
drives  of  this  type  can  be  made  more  efficient  than  is  possible 
with  either  worm  or  spiral  gears. 

Methods  of  Transmitting  Power  between  Non-parallel,  Non- 
intersecting  Shafts.  —  The  problem  of  transmitting  rotary 
motion  to  non-parallel,  non-intersecting  shafts  by  means  of 
toothed  wheels  has  been  solved  in  several  ways,  viz.,  by  spiral 
gears,  by  several  types  of  worm-gears,  and  by  skew  bevel  gears. 
While  spiral  gears  possess  the  advantage  of  working  perfectly 
when  the  center  distance  is  slightly  altered,  or  when  either  gear 
is  shifted  along  the  shaft,  they  have  the  disadvantage  of  rapid 
'  wear  on  account  of  point  contact.  A  worm  mating  with  a  helical 
gear  (straight  faced)  possesses  the  same  advantages,  but  is  also 
subject  to  the  same  disadvantages. 

A  straight  worm  mating  with  a  hobbed  wheel  (the  most  com- 
mon type)  has  line  contact,  and  consequently  better  wearing 
qualities  when  properly  mounted;  but  a  slight  displacement  of 
the  worm-wheel  from  its  correct  position  along  the  shaft  will 
seriously  affect  both  the  efficiency  and  durability  of  the  drive. 
The  Hindley  worm  and  wheel,  while  theoretically  giving  line 
contact  only,  are  so  constructed  that  to  the  eye  there  appears  to 
be  surface  contact.  When  properly  mounted  these,  too,  possess 
very  good  wearing  qualities,  but  the  slightest  displacement  of 
either  worm  or  wheel,  or  of  the  shafts,  will  affect  the  correct  work- 
ing of  the  gears. 

Types  of  Skew  Bevel  Gears.  —  To  students  of  gearing  prob- 
lems, the  solution  by  means  of  skew  bevel,  or  hyperboloidal  gears, 
has  always  seemed  the  logical  one,  although  up  to  the  present 
time  no  form  of  tooth  has  been  found  that  possesses  the  qualities 
of  strength,  reversibility,  and  low  pressure  angle,  and  that  can  be 
accurately  cut  or  planed.  Three  notable  attempts  to  produce 
such  gears  are  worthy  of  mention: 


SKEW  BEVEL  GEARS  293 

1.  The  epicycloidal  system  of  Willis,  in  which  the  tooth  sur- 
faces are  swept  out  by  an  element  of  a  hyperboloid  simultaneously 
rolling  on  the  interior  of  one  of  the  two  pitch  hyperboloids,  and 
on  the  exterior  of  the  other.     This  method  of  generating  the 
teeth  is  exactly  analogous  to  that  used  in  connection  with  the 
epicycloidal  system  in  spur  gearing,  but  Prof.  MacCord  has  shown 
that  skew  teeth  produced  in  this  way  are  not  theoretically  correct, 
since  the  tooth  surfaces  cannot  be  tangent  to  one  another  for  all 
positions. 

2.  The  Olivier  involute  system,  in  which  the  tooth  surfaces  are 
single  curved  surfaces  known  as  Olivier  spiraloids,  or  helical  con- 
volutes.     In  this  system,  the  teeth  vanish  at  the  gorge,  and  hence 
it  is  necessary  to  form  the  gears  from  sections  some  distance  from 
the  gorge,  but  as  we  depart  from  it,  the  obliquity  of  action  in- 
creases so  rapidly  that  it  is  difficult  to  find  a  position  where  the 
teeth  are  satisfactory.     Moreover,  this  method  provides  for  the 
generation  of  only  one  side  of  the  teeth.    The  backs  must  be 
formed  by  another  method. 

3.  The  Beale  skew  gear  is  a  modified  form  of  the  Olivier  gear, 
but  is  more  practical,  since  the  teeth  do  not  vanish  at  the  gorge. 
They  are  somewhat  undercut,  however,  and  like  all  the  preceding 
forms  are  difficult  to  make. 

It  does  not  seem  likely  that  any  type  of  skew  gear  will  ever 
be  of  more  than  theoretical  interest,  unless  the  teeth  are  of  such 
a  form  that  they  can  be  accurately  generated  by  the  planing 
process.  To  discover  this  form  of  tooth,  the  natural  method  of 
procedure  is  to  seek  something  that  will  bear  the  same  relation 
to  the  hyperboloidal  gear  that  the  rack  bears  to  the  involute  spur 
gear,  or  that  the  crown  gear  bears  to  the  octoidal  bevel  gear. 
Then,  assuming  that  Sang's  theory  is  applicable  to  skew  gears 
as  well  as  to  spur  and  bevel  gears,  a  tooth  section  with  straight 
sides  can  be  chosen  for  this  skew  rack  similar  to  those  of  the  spur 
and  bevel  gear  systems,  and  the  teeth  can  be  swept  out  by  a 
straight-edged  planing  tool  that  is  made  to  move  in  accordance 
with  the  proper  geometric  laws.  It  will  then  be  true  that  all 
hyperboloidal  gears  that  will  mesh  with  this  skew  rack  will  mesh 
with  each  other. 


294 


BEVEL  GEARING 


The  process  of  generating  the  teeth  of  a  pair  of  hyperboloidal 
gears  will  then  at  once  suggest  itself,  for  we  have  only  to  cause 
our  planing  tool  to  pass  along  the  tooth  surface  of  an  imaginary 
skew  rack  with  a  reciprocating  motion,  at  the  same  time  moving 
laterally  at  the  end  of  each  stroke  to  correspond  with  the  motion 


JfacMnenj 


Fig.  i.    Theory  of  Skew  Bevel  Gears  and  Requirements  for  Planing 
Conjugate  Teeth 

of  the  pitch  surface  of  the  rack  as  it  rolls  upon  the  pitch  hyper- 
boloid.  The  blank  must  also  turn  on  its  axis  a  corresponding 
amount  at  the  end  of  each  stroke,  these  motions  continuing  until 
one  side  of  a  tooth  has  been  completed. 

Solution  of  the  Problem.  —  The  nature  of  the  skew  rack  is 
determined  as  follows:   Let  AB,  Fig.  i,  be  the  axis  of  a  hyper- 


SKEW  BEVEL   GEARS  295 

boloid  of  revolution  of  one  nappe,  assumed  as  the  pitch  surface 
of  a  skew  bevel  gear.  The  horizontal  projection  of  the  circle  of 
the  gorge  is  cod,  and  the  vertical  projection,  c'o'd'  '.  Line  i'f  is 
one  of  the  rectilinear  elements  of  the  surface,  and  is  parallel  to 
the  plane  on  which  the  figure  is  projected.  In  this  position  it  is 
the  asymptote  of  the  hyperbola  which  constitutes  the  contour 
lines  of  the  surface.  The  angle  a  is  the  asymptote  angle  of  the 
hyperbola,  o'c'  is  the  radius  of  the  gorge  circle,  and  will  be 
referred  to  as  r;  c'G  parallel  to  AB,  will  be  referred  to  as  b.  The 
equation  of  the  hyperbola  referred  to  the  axes  AB  and  c'd'  '  , 
origin  at  of,  is: 


It  can  easily  be  proved  that  any  two  hyperboloids,  so  related 
that  they  can  form  the  pitch  surfaces  of  skew  gears,  will  have 
the  same  value  of  b.  Now  b  =  r  cot  a.  Hence  in  a  set  of  in- 
terchangeable hyperboloidal  gears,  the  product  of  the  gorge 
radius  and  the  cotangent  of  the  asymptote  angle  is  constant 
and  equal  to  b,  the  modulus  of  the  series.  Through  o,  the  in- 
tersection of  i'f  with  the  gorge  circle,  pass  a  line  XY,  perpen- 
dicular to  i'f  \  and  parallel  to  the  vertical  plane  of  projection. 
At  any  two  points  on  the  element  i'j'  equally  distant  from  o, 
such  as  V  and  /',  draw  lines  perpendicular  to  i'f  and  tangent  to 
the  hyberboloid.  The  slope  of  these  tangents  with  respect  to  a 

plane  perpendicular  to  XY  will  be  -7-7  .     Through  i'  and  /'  pass 

two  helices  the  common  axis  of  which  is  XY,  and  whose  slope  is 
the  same  as  that  of  the  tangents  through  i*  and/'.  These  helices 
will  then  be  tangent  to  the  hyperboloid.  Their  pitch  will  be  2  irb. 
Now  let  these  two  helices,  and  the  line  XY,  be  taken  as  the 
three  directrices  of  a  warped  surface.  The  result  will  be  a  right 
helicoid,  all  elements  of  which  will  be  perpendicular  to  XY.  One 
of  these  elements  is  evidently  i'f  .  This  helicoid  is  tangent  to 
the  hyperboloid  all  along  the  element  i'f  (since  three  common 
tangent  planes  can  be  passed  at  points  on  this  element).  If  the 
helicoid  is  moved  in  the  direction  of  XY,  it  will  roll  upon  the 
hyperboloid  while  turning  upon  its  own  axis,  and  successive 


2Q6 


BEVEL  GEARING 


Fig.  2. 


Pitch  Surfaces  of  Skew  Bevel  Gears  showing  Rectilinear 
Elements  of  Engagement 


elements  of  the  two  surfaces  will  come  in  contact.     The  sliding 
motion  that  accompanies  the  rolling  will  be  entirely  in  the  direc- 
tion of  the  element  of  contact. 
If  we  conceive  the  mate  to  the  hyperboloid  of  Fig.  i  to  be 


SKEW  BEVEL  GEARS  297 

placed  in  contact  with  it  along  the  element  IF  in  Fig.  2,  and  a 
right  helicoid  to  be  constructed  tangent  to  it  along  IF,  this 
helicoid  will  be  identical  with  the  first  one.  It  is  then  evident 
that  the  analogue  of  the  pitch  surface  of  the  rack,  for  hyper- 
boloidal  gears,  is  a  right  helicoid  the  pitch  of  which  is  2-n-b.  Now 
conceive  a  number  of  normals  to  the  hyperboloid  of  Fig.  i  erected 
at  points  of  the  common  element  i'f  or  IF  in  Fig.  2.  Each  will 
cut  AB,  and  also  the  axis  of  the  mating  hyperboloid.  Each  will 
be  parallel  to  any  plane  perpendicular  to  IF.  Hence  these  nor- 
mals are  elements  of  a  hyperbolic  paraboloid  which  is  normal,  not 
only  to  the  two  hyperboloids,  but  also  to  the  right  helicoid.  By 
laying  off  equal  distances  from  /  and  F  along  the  normals  through 
those  points,  and  joining  them  by  straight  lines,  the  second  gen- 
eration of  the  hyperbolic  paraboloid  will  be  formed,  the  plane 
director  of  which  is  parallel  to  IF  and  XY. 

Divide  the  axis  XY  of  the  helicoid  into  parts,  each  equal  to 
the  thickness  of  the  proposed  rack  tooth,  and  through  the  several 
points  of  division  pass  rectilinear  elements  of  the  helicoid.  Then 
through  each  of  these  elements  pass  a  hyperbolic  paraboloid 
normal  to  the  helicoid  (in  the  same  way  that  the  first  one  was 
passed  through  IF).  Now,  beginning  at  one  end  of  the  series, 
turn  the  first  hyperbolic  paraboloid,  about  the  helicoidal  element 
as  an  axis,  through,  say  15  degrees.  Turn  the  next  one  through 
the  same  angle  in  the  opposite  direction,  and  thus  alternate 
throughout  the  series.  A  limited  portion  of  these  surfaces, 
extending  equal  distances  on  both  sides  of  the  helicoid,  will  form 
the  surfaces  of  rack  teeth  which  are  analogous  in  every  respect 
to  the  rack  teeth  in  the  involute  spur  gear  system.  They  fulfill 
the  condition  required  by  Sang's  theory  for  a  conjugating  rack, 
viz.y  that  the  faces  and  flanks  be  formed  of  four  equal  curves 
arranged  in  alternate  reversion.  Hence  any  two  hyperboloidal 
gears  that  will  mesh  properly  with  this  rack  will  mesh  with  each 
other.  Three  of  these  rack  teeth  are  shown  in  Fig.  i. 

If  we  imagine  a  planing  tool  the  point  of  which  is  made  to 
travel  along  the  straight  line  at  the  root  of  the  rack  tooth,  and 
whose  cutting  edge,  at  the  same  time,  is  made  to  turn  in  such  a 
way  that  it  constantly  coincides  with  some  rectilinear  element 


2p8  BEVEL   GEARING 

of  the  tooth  surface  (see  the  tooth  at  JN,  Fig.  i),  it  only  remains 
for  us  to  supply  the  mechanism  necessary  to  give  the  proper 
motions  to  the  hyperboloidal  blank  and  to  the  imaginary  heli- 
coidal  rack,  and  we  shall  have  a  means  of  accurately  generating 
conjugate  teeth  for  skew  gears.  The  teeth  in  these  gears  do  not 
vanish  at  the  gorge,  as  do  those  of  Olivier  and  Herrmann.  The 
obliquity  of  action  is  the  same  at  points  some  distance  from  the 
gorge  as  it  is  at  the  gorge.  The  teeth  do  not  have  to  be  undercut 
to  make  them  work  at  the  gorge;  hence  they  are  not  weakened 
near  this  point.  The  gears  are  reversible,  and  by  using  sections 
of  the  hyperboloids  some  distance  from  the  gorge,  drives  of  this 
type  can  be  made  more  efficient  than  either  worm  or  spiral  gears. 
Practical  Application  of  Theory.  —  Gears  produced  according 
to  the  theoretical  principle  just  outlined  have  been  cut  by  Mr. 
Max  Uhlmann,  of  Philadelphia,  who  states  that  the  theoretical 
considerations  of  Bartlett  are  proved  to  be  correct  by  the  carry- 
ing out  of  the  work  in  practice,  except  in  regard  to  that  part  of 
the  theory  which  relates  to  under-cutting.  Owing  to  the  length- 
wise sliding  or  slip  —  if  it  may  be  so  called  —  it  might  be  ex- 
pected that  the  tooth  outline  of  the  skew  gear  would  differ  from 
that  of  the  corresponding  bevel  gear,  and  to  bring  out  its  peculiar 
features  as  clearly  as  possible,  Mr.  Uhlmann  made  a  pair  of  gears 
which  extended  as  near  to  the  gorge  circle  of  the  hyperboloid  as 
could  be  conveniently  cut  with  his  present  facilities.  The  di- 
mensions of  these  gears  were  as  follows: 

Largest  pitch  diameter  =  2.649  inches. 

Angle  of  asymptote  =  30  degrees. 

Number  of  teeth  =  12. 

Width  of  gear  face  =  1.25  inch. 

Diameter  of  gorge  circle  =  0.875  inch. 

From  this  data  the  pitch  of  the  imaginary  helicoidal  rack 
figures  out  4.761  inches. 

The  hyperboloidal  form  of  the  blanks  was  turned  with  the  tool 
set  half  an  inch  above  the  center,  and  with  the  compound  rest 
set  to  the  same  angle  at  which  it  would  have  been  set  for  turning 
the  corresponding  bevel  gear  blank.  The  method  of  turning 
the  blanks  will  be  readily  understood  by  reference  to  Fig.  3. 


SKEW  BEVEL   GEARS 


299 


Although  a  pressure  angle  of  20  degrees  was  used,  there  is  not 
only  a  considerable  under-cutting  —  which  is  all  on  one  side  of 
the  tooth  —  but  the  tool  also  cuts  away  material  on  the  same  side 
of  the  tooth  at  the  top;  the  effect  on  the  opposite  side  of  the  tooth 
is  just  the  reverse  so  that  it  will  be  apparent  that  the  tooth  out- 
line is  decidedly  unsymmetrical.  From  the  study  made  of  this 
subject,  it  is  believed  that  this  deviation  from  a  symmetrical 
tooth  outline  will  be  most  pronounced  at  some  point  near  the 


SKEW  GEAR  BLANK 


BEVEL  GEAR  BLANK  SKEW  GEAR 

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Fig.  3.    Turning  the  Blanks  for  Skew  Bevel  Gears  and  Regular  Bevel  Gears 

gorge  circle,  and  that  beyond  the  gorge  circle  these  conditions 
reverse  so  that  excessive  under-cutting  will  then  appear  on  the 
opposite  side  of  the  tooth. 

In  one  case  of  skew  bevel  gears  this  excessive  under-cutting 
has  been  avoided  by  changing  the  pressure  angle  on  one  side  to 
25  degrees,  and  simultaneously  changing  the  pressure  angle  on 
the  other  side  to  15  degrees.  This  produces  a  somewhat  "lean- 
ing tooth,"  but  what  under-cutting  there  is  on  each  side  is  only 
trifling  in  its  amount. 


300  BEVEL   GEARING 

Skew  bevel  gears  find  application  in  various  kinds  of  machines, 
but  nearly  all  of  these  gears  are  now  cast  from  patterns,  while 
even  the  best  of  cut  gears  of  this  type,  which  the  trade  can  so  far 
supply,  are  far  from  being  good  approximations.  The  method 
of  generating  teeth  here  outlined,  should  afford  a  means  of  pro- 
ducing skew  bevel  gears  which,  in  point  of  accuracy,  and  conse- 
quently smooth  action,  would  be  equal  to  generated  bevel  gears. 
Where  the  distance  between  the  shafts  is  great  enough  to  make 
the  gorge  circle  very  large,  it  will  usually  be  possible  to  use  other 
forms  of  gears;  but  when  this  distance  is  only  great  enough  to 
allow  the  shafts  to  pass  each  other,  thus  requiring  the  gorge 
circle  to  be  small,  as  is  frequently  the  case,  skew  bevel  gears  are 
the  only  form  of  transmission  which  will  answer  the  purpose. 


INDEX 


Acute-angle  bevel  gearing,  213,  218 
Addendum,  10,  16 

angular,  204 

corrected,  33 

for   diametral,    circular   and    metric 
pitches,  288 

gears  with  long  and  short,  122,  278, 
282 

table,  22,  23 
Addendum  angle,  204 
Angle,  addendum,  204 

center,  200 

cutting,  204 

dedendum,  204 

edge,  204 

face,  204 

of  pressure  or  obliquity,  9,  85 

pitch  cone,  201 
Angular  addendum,  204 
Annealing  steel  for  gears,  171 
Automobile    transmission    gears,    pro- 
duction of,  167 

JJeale  skew  bevel  gear  teeth,  293 

Bearing  pressures  of  bevel  gears,  231 
Bevel  gears,  200 

acute-angle,  213,  218 

bearing  pressures,  231 

blanks,  236 

calculations,  derivation  of  formulas, 
207 

calculations,  examples,  212 

crown,  203,  215,  219 

cutting  the  teeth  in,  243,  261,  274 

definitions,  200,  202 

design,  224,  236 

drawings  for,  229,  238 

generating  machines  for,  258 

horsepower  transmitted,  227,  230 


Bevel  gears,  internal,  203,  216,  219 

internal,  how  to  avoid,  220 

materials  for,  224 

milling  the  teeth,  261,  274 

milling  the  teeth,  table  of  "set-over," 
266 

miter,  212 

obtuse-angle,  214,  218 

planing,  adjustable  former  for,  255 

right-angle,  210,  212 

rules  and  formulas,  200,  210 

simplified  formulas  for  strength,  230 

skew,  291 

strength,  224 

teeth,  cutters  for,  222 

teeth,  different  systems  for,  220 

teeth,  filing,  269 

teeth,  machines  for  cutting,  253 

teeth,  special  forms,  223 

teeth,  table  of  "set-over"  when  mill- 
ing, 266 

with  long  and  short  addendum,  127, 
278,  282 

with  long  and  short  addendum,  dupli- 
cation of,  284 

with     planed     teeth,     dimensioning 

drawings  for,  241 
Blank  diameters,  limits  for,  36 
Blanks  for  bevel  gears,  236 
Brass  gears,  43 

Broaching  automobile  gears,  172 
Bronze  gears,  43,  224 

Casehardening  gears,  49,  60,  175,  177, 

184,  195 
Cast-iron  gears,  38,  40,  47,  224 

working  stresses,  47,  52 
Cast-steel  gears,  38,  40,  225 
Center  angle  in  bevel  gearing,  200 


301 


302 


CENTER  —  FORMED 


Center  distance,  10,  15 
manufacturing  limits  for,  36 

Chart  for  power  transmission  by  gear- 
ing, 82 

Chordal  pitch,  10 

Chordal  thickness,  10,  33 

gears  with  long  addendum,  279 

Circular  and  diametral  pitch,  table,  22, 

23 

Circular  pitch,  10,  12 
Circular  pitch  tooth  parts,  23 
Clearance,  10,  17 

of  gears  cut  on  gear  shaper,  17 
Cloth  gears,  44 
Corrected  addendum,  33 
Cost  of  heat-treated  gears,  193 
Crown  gears,  203,  215,  219 
Curve,  cycloidal,  5 

involute,  4 
Cutters,  for  bevel  gear  teeth,  222 

for  spur  gear  teeth,  31 

grinding,  31 

Cutting  angle  in  bevel  gearing,  204 
Cutting  teeth,  in  automobile  gears,  1 74 

in  bevel  gears,  243,  261,  274 

in  spur  gears,  136 
Cycloidal  system  of  gear  teeth,  5,  85 

comparison  with  involute  systems,  6 

for  bevel  gearing,  221 

odontograph  for,  27 

used  for  racks,  6,  7 

Dedendum,  10 

angle,  204 

Definitions,  bevel  gearing,  200,  202 
Definitions  of  tooth  parts,  9 
Depth  of  tooth,  17,  22,  23 

diametral  and  circular  pitches,  22,  23 

metric  pitch,  288 

working,  10 
Describing-generating  principle  of  gear 

cutting,  140 
Design,  bevel  gears,  224,  236 

spur  gears,  129 
Device  for  testing  and  measuring  gears, 

198 
Diameter,  outside,  10,  18 

outside  of  blank,  limits  for,  36 


Diameter,  pitch,  10,  14 

pitch,  limits  for,  36 

root,  10 
Diametral  and  circular  pitch,  table,  22, 

23 
Diametral  pitch,  10,  12 

relation  to  width  of  face,  56 

tooth  parts,  22 

Diametral  pitches  in  full  size,  n 
Dimensions,  of  spur  gears,  21,  131 

*of  bevel  gears,  210 

on  bevel  gear  drawings,  229,  238,  241 

on  rack  drawings,  135 

on  spur  gear  drawings,  133 
Drawings,  for  bevel  gears,  229,  238 

for  bevel  gears  with  planed  teeth, 
dimensioning,  241 

for  spur  gears,  133,  134 

properly  dimensioned  rack,  135 
Drop-forgings  for  gears,  42,  170 
Duplication   of    gears   with   long   and 

short  addendum,  284 
Durability  of  gears,  65,  230 

Edge  angle  in  bevel  gearing,  204 

Efficiency  of  spur  gears,  66 
Epicycloidal  curve,  5 
Equivalent  spur  gear,  205 

pace  angle,  204 

Face  of  gear  tooth,  10 
Face  width,  relation  to  diametral  pitch, 

56 
Fellows  gear  shaper,  clearance  of  gears 

cut  on,  17 

Fellows  stub  gear-tooth,  84,  94 
Fiber  gears,  44,  225 
Filing  bevel  gear  teeth,  269 
Flank  of  gear  tooth,  10 
Flutes  in  hobs,  number  of,  152 

straight  or  spiral,  153 
Forgings  for  gears,  drop,  42,  170 
Formed  cutters,  for  bevel  gear  teeth,  222 
for  spur  gear  teeth,  31 
grinding,  31 
Formed  tool  principle,  for  bevel  gear 

cutting,  245 
for  spur  gear  cutting,  138 


FORMER  —  GEARS 


303 


Former  for  bevel  gear  planing,  adjust- 
able, 255 
Friction  wheels,  i 

Gear-cutters,  for  bevel  gears,  222 
for  spur  gears,  31 

grinding,  31 

Gear-cutting  machinery,  classification, 
136 

for  bevel  gears,  253,  258 

for  spur  gears,  145 
Gear  drawing,  model  bevel,  229,  238 

model  rack,  135 

model  spur,  133,  134 
Gear  shaper,  clearance  of  gears  cut  on,  1 7 
Gear  teeth,  comparison  of  involute  and 
cycloidal,  6 

cutters  for  milling,  31,  222 

cutting  bevel,  243,  261 

cutting  spur,  136 

cycloidal,  5,  85 

definitions  of  parts  of,  9 

dimensions,  full  size,  n 

easing  off  the  points,  74 

filing  bevel,  269 

for  bevel  gearing,  220 

bobbing,  147 

involute,  4,  85 

involute,  variations  in  tooth  shape, 
156 

machines  for  cutting  bevel,  253,  258 

machines  for  cutting  spur,  145 

metric  or  module  system,  36,  280,  288 

relation  of  strength  to  velocity,  67 

rolling  mill,  95 

skew  bevel,  293 

special  forms  for  bevel,  223 

strength  of  bevel,  225 

strength  of  spur,  51 

strength  of  stub,  96,  99 

stub,  84 

stub  and  standard  compared,  92 

tables  of  parts  of,  22,  23 

tests  of  strength,  59 
Gears,  bevel,  200 

bevel,  cutting  teeth  in,  243,  261,  274 

bevel,  horsepower  transmitted,   227, 
230 


Gears,  bevel,  materials  for,  224 
bevel,  strength  and  design,  224,  236 
brass,  43 
bronze,  43,  224 
cast  iron,  38,  40,  47,  52,  224 
cast  steel,  38,  40,  225 
cloth,  44 

comparison  of  heat-treatments,  184 
crown,  203,  215,  219 
durability,  65,  230 
efficiency,  66 

equipment  for  heat-treating,  189 
'fiber,  44,  225 
for  machine  tools,  46,  182 
for  rolling  mills,  95 
hardness  testing,  50,  180 
heat-treated,  cost,  193 
heat-treatment,    49,    60,    167,    175, 

195 

interchangeability  of  hobbed  and 
milled,  155 

internal  bevel,  203,  216,  219 

internal  bevel,  how  to  avoid,  220 

internal  spur,  24 

limits  for,  36 

make-shift  methods  for  avoiding 
noisy,  127 

materials  used  for,  38  • 

methods  of  production,  167 

noisy,  108 

price,  83 

rawhide,  43,  62,  225 

rawhide,  horsepower  transmitted,  63 

running-in,  181 

simplified  formulas  for  strength,  75 

skew  bevel,  291 

spur,  design  of,  129 

spur,  device  for  testing  and  measur- 
ing, 198 

spur,  dimensions,  21,  131 

spur,  horsepower  transmitted,  53 

spur,  principles  and  dimensions,  i 

spur,  rules  and  formulas  for,  21 

steel,  42,  47,  224 

steel  used  for,  49,  60 

strength  of,  51,  224 

stub-tooth,  84 

weight,  83 


304 


GEARS  —  MOLDING 


Gears,  with  long  and  short  addendum, 

122,  278,  282 

with  long  and  short  addendum,  dupli- 
cation of,  284 

Generating  hob-tooth  shapes,  160 
Generating  machines  for  bevel  gears, 

258 
Generating  principle,  for  cutting  bevel 

gears,  248 

for  cutting  skew  bevel  gears,  291 
for  cutting  spur  gears,  140,  142 
Grant's  odontograph,  27,  28,  30 
Grinding  gear  cutters,  31 

Hardening  gears,  49,  60,  175,  184,  195 

Hardening  hobs,  154 
Hardening  methods,  195 
Hardness  testing  of  gears,  50,  180 
Heat-treated  gears,  cost,  193 
Heat-treatment  of  gears,  49,  60,  167 

equipment  for,  189 

for  machine  tools,  182 

methods,  175,  194 
Heat-treatment  of  hobs,  154 
Hob  teeth,  form  and  dimensions,  150 

generating,  160 

relief,  151,  154 
Hobs  for  spur  gears,  designing,  150 

heat- treatment,  154 

master  planing  tool  for,  164 

number  of  flutes,  152 

special,  158 

straight  or  spiral  flutes,  153 

threading,  153 

Hobbed  and  milled  gears,  interchange- 
ability  of,  155 
Robbing  gear  teeth,  147 
Horsepower  transmitted,  by  bevel  gear- 
ing, 227,  230 

by  spur  gearing,  53,  63,  82 
Hypocycloidal  curve,  5 

Impact,  variation  in  strength  due  to, 

68,70 
Interchangeability  of  hobbed  and  milled 

gears,  155 
Interference,  a  cause  of  noisy  gearing, 

no,  112 


Interference,  in  internal  gears,  26 
Internal  bevel  gears,  203,  216,  219 

how  to  avoid,  220 
Internal  spur  gears,  24 

chart  for  number  of  teeth,  121 

interference,  26 

rules  for  designing,  25 

with  shortened  addendum,  125 
Involute  system  of  gear  teeth,  4,  85 

comparison  with  cycloidal  system,  6 

for  bevel  gearing,  221 

for  racks,  6,  7,  30 

odontograph  for,  30 

variations  in  tooth  shape,  156 

Lewis  formula,  51,  227 

derivation,  57 
factors  for  bevel  gears,  226 
factors  for  spur  gears,  54 
factors  for  stub-tooth  gears,  106 
Limits  for  gearing,  36 
Long  and  short  addendum  gears,  122, 
278,  282 


tool  gears,  46,  182 
Machines,    classification    of    gear 

cutting,  136 

for  cutting  bevel  gear  teeth,  253,  258 
for  cutting  spur  gear  teeth,  145 
Materials,  for  bevel  gears,  224 
for  racks,  45 
for  spur  gears,  38 
Measuring  gear  teeth,  35 
Measuring  gears,  device  for,  198 
Metric  system  of  gear  teeth,  36,  280,  288 
Milling  bevel  gears,  261,  274 
method  of  setting  cutter,  274 
table  of  "  set-over,"  266 
Milling  cutters,  for  bevel  gear  teeth,  222 
for  gear  teeth,  grinding,  31 
for  spur  gear  teeth,  3  1 
Miter  bevel  gearing,  212 
Module  system  of  gear  teeth,  36,  280, 

288 
Molding-generating  principle,  for  cut- 

ting bevel  gears,  248 
for  cutting  spur  gears,  142 
for  producing  skew  bevel  gear  teeth, 
291 


NOISY  —  SPUR 


305 


Noisy  gearing,  108 

make-shift  methods  for  avoiding, 

127 

Number  of  teeth  in  gears,  19,  21 
Nuttall  system  of  stub  gear-teeth,  94 

Obliquity  angle,  9,  85 

Obtuse-angle  bevel  gearing,   214, 

218 

Octoid  teeth  for  bevel  gearing,  221 
Odontograph,  for  cycloidal  teeth,  27 

for  involute  teeth,  30 
Odontographic    principle,    for    cutting 
bevel  gears,  247 

for  cutting  spur  gears,  139 
Oil  for  quenching  gears,  192 
Oil-hardened  gears,  184 
Olivier  system  for  skew  bevel  gear  teeth, 

293 
Phosphor-bronze  gears,  working  stresses, 

52 
Pitch,  chordal,  10 

circular  and  diametral,  10,  12,  22,  23 

diametral,  in  full  size,  n 

diametral,  relation  to  width  of  face,  56 
Pitch  circle,  10 
Pitch  cone,  200 
Pitch  cone  angle,  201 
Pitch  cone  radius,  204 
Pitch  cylinders,  3 
Pitch  diameter,  4,  10,  14 

in  bevel  gearing,  202 

manufacturing  limits  for,  36 
Pitch  line,  3 
Planing  bevel  gears,  250 

adjustable  former  for,  255 
Planing  tools  for  hobs,  164 
Power  transmitted,  by  bevel  gearing, 
227,  230 

by  spur  gearing,  51,  53,  82 
Pressure  angle,  9,  85 
Pressure  on  bearings,  bevel  gears,  231 
Price  of  gears,  83 
Production  of  gears,  167 

Quenching  oil  for  gears,  192 

Rack  and  pinion  with  shortened  adden- 
dum, 125 


Racks,  materials  for,  45 

properly  dimensioned  drawing  for,  135 

straightening,  46 

teeth,  cycloidal,  6,  7 

teeth,  involute,  7,  30 
Rawhide  gears,  43,  225 

allowable  load,  62,  65 

power  transmitted,  63 

strength,  62 

with  flanges,  64 
Relief  of  hob  teeth,  151,  154 
Right-angle  bevel  gearing,  210,  212 
Rolling  mill  gears,  95 
Root  diameter,  10 
Running-in  of  gears,  181 

Sang's  theory,  291 

Scleroscope  test  for  gears,  50 
Shaping  bevel  gear  teeth,  250 
Shocks,  variation  in  strength  due  to,  68, 

70 

Short  addendum  gears,  122,  278,  282 
Skew  bevel  gears,  291 
Spur  gear,  equivalent,  205 
Spur  gears,  i 

cutting  teeth  in,  136 

device  for  testing  and  measuring,  198 

design,  129 

dimensions,  12,  21,  131 

drawings  for,  133,  134 

durability,  65 

efficiency,  66 

for  machine  tools,  46,  182 

hardness  testing,  50,  180 

heat-treated,  cost,  193 

heat-treatment,  49,  60,  167,  194 

hobbed  and  milled,  147,  155 

horsepower  transmitted,  53 

internal,  24 

machines  for  cutting,  145 

make-shift     methods     for     avoiding 
noisy,  127 

materials  used  for,  38 

noisy,  108 

price,  83 

principles,  i 

production,  167 

rules  and  formulas,  21 


306 


SPUR  —  WORKING 


Spur  gears,  running-in  of,  181 

strength,  51,  75 

stub-tooth,  84 

weight,  83 

with  long  and  short  addendum,  122, 

279 

Steel  castings  for  gears,  38,  40,  225 
Steel,  for  automobile  gears,  168 

for  gears  in  machine  tools,  183 

for  spur  gears,  49,  60 
Steel  gears,  working  stresses,  52 
Straightening  racks,  46 
Strength  of  bevel  gears,  224 
Strength  of  gear  teeth,  51,  225 

relation  to  velocity,  67 

simplified  formulas,  75 

stub-tooth,  96,  99 

tests,  59 
Stub-tooth  gears,  84 

advantages,  101 

compared  with  standard,  92 

dimensions,  94 

factors  for  Lewis  formula,  106 

strength,  96,  99 

Teeth,  chordal  thickness,  10,  33 

comparison  of  involute  and  cy- 
cloidal,  6 

cutters  for  bevel  gear,  222 
cutters  for  spur  gear,  31 
cutting  bevel  gear,  243,  261 
cutting,  in  automobile  gears,  174 
cutting  spur  gear,  136 
cycloidal,  5,  85 

cycloidal,  for  bevel  gearing,  221 
depth,  17,  22,  23 
easing  off  the  points,  74 
filing  bevel  gear,  269 
for  bevel  gearing,  systems  of,  220 
hob,  form  and  dimensions,  150 
hobbing,  147 
involute,  4,  85 

involute,  for  bevel  gearing,  221 
involute  rack,  30 
involute,  variations  in  tooth  shape, 

156 
machines  for  cutting  bevel  gear,  253, 

258 


Teeth,  machines  for  cutting  spur  gear, 

J45 

measuring  gear,  35 

metric  or  module  system,  36,  280,  288 

milling  the  teeth  of  bevel,  261,  274 

number  of,  19,  21 

octoid,  for  bevel  gearing,  221 

odontograph  for  laying  out,  27,  30 

rack,  6,  7 

relation  of  strength  to  velocity,  67 

shown  in  full  size,  n 

skew  bevel  gear,  293 

special  forms  for  bevel  gear,  223 

strength,  51,  225 

strength  of  stub  gear,  96,  99 

stub  gear,  84 

stub  gear,  and  standard  compared,  92 

tests  of  strength,  59 

thickness  of,  10,  18,  22,  23 

with  shortened  addendum,  122,  278 
Templet   principle,    for   cutting   bevel 
gears,  246 

for  cutting  spur  gears,  138 
Templets  for  gear  hobs,  161 
Testing  gears,  device  for,  198 
Testing  hardness  of  gears,  50,  180 
Thickness  of  tooth,  10,  18,  22,  23 

chordal,  10,  33 

chordal,  for  gears  with  long  adden- 

dum, 279 

Threading  hobs,  153 
Thrust  on  bearings,  bevel  gears,  231 
Tooth  depth,  for  diametral,  circular  and 

metric  pitches,  288 
Tooth  parts,  definitions,  9 

tables,  22,  23 

Tooth-shapes  for  hobs,  generating,  160 
Tooth  thickness,  10,  18,  22,  23 
Transmission  gears,  production  of  au- 

tomobile, 167 
Tredgold's  approximation,  244 

\e\ocity,  relation  to  strength  of  gear 

teeth,  67 
Vertex  distance,  204 


of  gears,  83 
Willis  skew  bevel  gear  teeth,  293 
Working  depth  of  teeth,  10,  22,  23 


GENERAL  LIBRARY 
UNIVERSITY  OF  CALIFORNIA— BERKELEY 

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OCTll  1954  LO 


REC'D  LD 

DEC  3 1  1957 

Apr  '60BM 
RFIC'D  LD 


1  May  '621V 

LD; 


,'UN    71962 


LD  21-100?n-l,'54(1887sl6)476 


JUN  2  8  1980 


JUN  2  8  2003 


.416066 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


